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Special Angles Kids: Sum and Difference Formulas

Special Angles Kids are those angles whose trigonometric functions can be found based on the trigonometric formulas of special angles.

Sum Formulas
$(1.)\;\; \sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \\[3ex] (2.)\;\; \cos (\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \\[3ex] (3.)\;\; \tan (\alpha + \beta) = \dfrac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \\[5ex]$ Difference Formulas
$(1.)\;\; \sin (\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta \\[3ex] (2.)\;\; \cos (\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \\[3ex] (3.)\;\; \tan (\alpha - \beta) = \dfrac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta} \\[5ex]$ Determine the trigonometric functions of these angles
Show all work

$75^\circ$
$75^\circ = 45^\circ + 30^\circ$
Sum Formula

$\sin 75 = \sin(45 + 30) \\[3ex] = \sin45\cos30 + \cos45\sin30 \\[3ex] = \dfrac{\sqrt{2}}{2} * \dfrac{\sqrt{3}}{2} + \dfrac{\sqrt{2}}{2} * \dfrac{1}{2} \\[5ex] = \dfrac{\sqrt{2 * 3}}{2 * 2} + \dfrac{\sqrt{2} * 1 }{2 * 2} \\[5ex] = \dfrac{\sqrt{6}}{4} + \dfrac{\sqrt{2}}{4} \\[5ex] \sin 75^\circ = \dfrac{\sqrt{6} + \sqrt{2}}{4}$

$\cos 75 = \cos(45 + 30) \\[3ex] = \cos45\cos30 - \sin45\sin30 \\[3ex] = \dfrac{\sqrt{2}}{2} * \dfrac{\sqrt{3}}{2} - \dfrac{\sqrt{2}}{2} * \dfrac{1}{2} \\[5ex] = \dfrac{\sqrt{2 * 3}}{2 * 2} - \dfrac{\sqrt{2} * 1 }{2 * 2} \\[5ex] = \dfrac{\sqrt{6}}{4} - \dfrac{\sqrt{2}}{4} \\[5ex] \cos 75^\circ = \dfrac{\sqrt{6} - \sqrt{2}}{4}$

$\tan 75 = \tan(45 + 30) \\[3ex] = \dfrac{\tan45 + \tan30}{1 - \tan45\tan30} \\[5ex] = \left(1 + \dfrac{\sqrt{3}}{3}\right) \div \left(1 - 1 * \dfrac{\sqrt{3}}{3}\right) \\[5ex] = \left(\dfrac{3}{3} + \dfrac{\sqrt{3}}{3}\right) \div \left(1 - \dfrac{\sqrt{3}}{3}\right) \\[5ex] = \left(\dfrac{3 + \sqrt{3}}{3}\right) \div \left(\dfrac{3}{3} - \dfrac{\sqrt{3}}{3}\right) \\[5ex] = \dfrac{3 + \sqrt{3}}{3} \div \dfrac{3 - \sqrt{3}}{3} \\[5ex] = \dfrac{3 + \sqrt{3}}{3} * \dfrac{3}{3 - \sqrt{3}} \\[5ex] = \dfrac{3 + \sqrt{3}}{3 - \sqrt{3}} \\[5ex] = \dfrac{3 + \sqrt{3}}{3 - \sqrt{3}} * \dfrac{3 + \sqrt{3}}{3 + \sqrt{3}} \\[5ex] = \dfrac{(3 + \sqrt{3})(3 + \sqrt{3})}{(3 - \sqrt{3})(3 + \sqrt{3})} \\[5ex] = \dfrac{9 + 3\sqrt{3} + 3\sqrt{3} + (\sqrt{3})^2}{3^2 - (\sqrt{3})^2} \\[5ex] = \dfrac{9 + 6\sqrt{3} + 3}{9 - 3} \\[5ex] = \dfrac{12 + 6\sqrt{3}}{6} \\[5ex] = \dfrac{6(2 + \sqrt{3})}{6} \\[5ex] \tan 75^\circ = 2 + \sqrt{3}$

$\csc 75 = \dfrac{1}{\sin 75} = 1 \div \sin75 \\[5ex] = 1 \div \dfrac{\sqrt{6} + \sqrt{2}}{4} \\[5ex] = 1 * \dfrac{4}{\sqrt{6} + \sqrt{2}} \\[5ex] = \dfrac{4}{\sqrt{6} + \sqrt{2}} * \dfrac{\sqrt{6} - \sqrt{2}}{\sqrt{6} - \sqrt{2}} \\[5ex] = \dfrac{4(\sqrt{6} - \sqrt{2})}{(\sqrt{6})^2 - (\sqrt{2})^2} \\[5ex] = \dfrac{4(\sqrt{6} - \sqrt{2})}{6 - 2} \\[5ex] = \dfrac{4(\sqrt{6} - \sqrt{2})}{4} \\[5ex] \csc 75^\circ = \sqrt{6} - \sqrt{2}$

$\sec 75 = \dfrac{1}{\cos 75} = 1 \div \cos75 \\[5ex] = 1 \div \dfrac{\sqrt{6} - \sqrt{2}}{4} \\[5ex] = 1 * \dfrac{4}{\sqrt{6} - \sqrt{2}} \\[5ex] = \dfrac{4}{\sqrt{6} - \sqrt{2}} * \dfrac{\sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}} \\[5ex] = \dfrac{4(\sqrt{6} + \sqrt{2})}{(\sqrt{6})^2 - (\sqrt{2})^2} \\[5ex] = \dfrac{4(\sqrt{6} + \sqrt{2})}{6 - 2} \\[5ex] = \dfrac{4(\sqrt{6} + \sqrt{2})}{4} \\[5ex] \sec 75^\circ = \sqrt{6} + \sqrt{2}$

$\cot 75 = \dfrac{1}{\tan 75} = 1 \div \tan75 \\[5ex] = 1 \div (2 + \sqrt{3}) \\[3ex] = \dfrac{1}{2 + \sqrt{3}} \\[5ex] = \dfrac{1}{2 + \sqrt{3}} * \dfrac{2 - \sqrt{3}}{2 - \sqrt{3}} \\[5ex] = \dfrac{2 - \sqrt{3}}{(2)^2 - (\sqrt{3})^2} \\[5ex] = \dfrac{2 - \sqrt{3}}{4 - 3} \\[5ex] = \dfrac{2 - \sqrt{3}}{1} \\[5ex] \cot 75^\circ = 2 - \sqrt{3} \\[3ex]$

$15^\circ$
$15^\circ = 45^\circ - 30^\circ$
Difference Formula

$\sin 15 = \sin(45 - 30) \\[3ex] = \sin45\cos30 - \cos45\sin30 \\[3ex] = \dfrac{\sqrt{2}}{2} * \dfrac{\sqrt{3}}{2} - \dfrac{\sqrt{2}}{2} * \dfrac{1}{2} \\[5ex] = \dfrac{\sqrt{2 * 3}}{2 * 2} - \dfrac{\sqrt{2} * 1}{2 * 2} \\[5ex] = \dfrac{\sqrt{6}}{4} - \dfrac{\sqrt{2}}{4} \\[5ex] \sin 15^\circ = \dfrac{\sqrt{6} - \sqrt{2}}{4}$

$\cos 15 = \cos(45 - 30) \\[3ex] = \cos45\cos30 + \sin45\sin30 \\[3ex] = \dfrac{\sqrt{2}}{2} * \dfrac{\sqrt{3}}{2} + \dfrac{\sqrt{2}}{2} * \dfrac{1}{2} \\[5ex] = \dfrac{\sqrt{2 * 3}}{2 * 2} + \dfrac{\sqrt{2} * 1}{2 * 2} \\[5ex] = \dfrac{\sqrt{6}}{4} + \dfrac{\sqrt{2}}{4} \\[5ex] \cos 15^\circ = \dfrac{\sqrt{6} + \sqrt{2}}{4}$

$\tan 15 = \tan(45 - 30) \\[3ex] = \dfrac{\tan45 - \tan30}{1 + \tan45\tan30} \\[5ex] = \left(1 - \dfrac{\sqrt{3}}{3}\right) \div \left(1 + 1 * \dfrac{\sqrt{3}}{3}\right) \\[5ex] = \left(\dfrac{3}{3} - \dfrac{\sqrt{3}}{3}\right) \div \left(1 + \dfrac{\sqrt{3}}{3}\right) \\[5ex] = \left(\dfrac{3 - \sqrt{3}}{3}\right) \div \left(\dfrac{3}{3} + \dfrac{\sqrt{3}}{3}\right) \\[5ex] = \dfrac{3 - \sqrt{3}}{3} \div \dfrac{3 + \sqrt{3}}{3} \\[5ex] = \dfrac{3 - \sqrt{3}}{3} * \dfrac{3}{3 + \sqrt{3}} \\[5ex] = \dfrac{3 - \sqrt{3}}{3 + \sqrt{3}} \\[5ex] = \dfrac{3 - \sqrt{3}}{3 + \sqrt{3}} * \dfrac{3 - \sqrt{3}}{3 - \sqrt{3}} \\[5ex] = \dfrac{(3 - \sqrt{3})(3 - \sqrt{3})}{(3 + \sqrt{3})(3 - \sqrt{3})} \\[5ex] = \dfrac{9 - 3\sqrt{3} - 3\sqrt{3} + (\sqrt{3})^2}{3^2 - (\sqrt{3})^2} \\[5ex] = \dfrac{9 - 6\sqrt{3} + 3}{9 - 3} \\[5ex] = \dfrac{12 - 6\sqrt{3}}{6} \\[5ex] = \dfrac{6(2 - \sqrt{3})}{6} \\[5ex] \tan 15^\circ = 2 - \sqrt{3}$

$\csc 15 = \dfrac{1}{\sin 15} = 1 \div \sin15 \\[5ex] = 1 \div \dfrac{\sqrt{6} - \sqrt{2}}{4} \\[5ex] = 1 * \dfrac{4}{\sqrt{6} - \sqrt{2}} \\[5ex] = \dfrac{4}{\sqrt{6} - \sqrt{2}} * \dfrac{\sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}} \\[5ex] = \dfrac{4(\sqrt{6} + \sqrt{2})}{(\sqrt{6})^2 - (\sqrt{2})^2} \\[5ex] = \dfrac{4(\sqrt{6} + \sqrt{2})}{6 - 2} \\[5ex] = \dfrac{4(\sqrt{6} + \sqrt{2})}{4} \\[5ex] \csc 15^\circ = \sqrt{6} + \sqrt{2}$

$\sec 15 = \dfrac{1}{\cos 15} = 1 \div \cos15 \\[5ex] = 1 \div \dfrac{\sqrt{6} + \sqrt{2}}{4} \\[5ex] = 1 * \dfrac{4}{\sqrt{6} + \sqrt{2}} \\[5ex] = \dfrac{4}{\sqrt{6} + \sqrt{2}} * \dfrac{\sqrt{6} - \sqrt{2}}{\sqrt{6} - \sqrt{2}} \\[5ex] = \dfrac{4(\sqrt{6} - \sqrt{2})}{(\sqrt{6})^2 - (\sqrt{2})^2} \\[5ex] = \dfrac{4(\sqrt{6} - \sqrt{2})}{6 - 2} \\[5ex] = \dfrac{4(\sqrt{6} - \sqrt{2})}{4} \\[5ex] \sec 15^\circ = \sqrt{6} - \sqrt{2}$

$\cot 15 = \dfrac{1}{\tan 15} = 1 \div \tan15 \\[5ex] = 1 \div (2 - \sqrt{3}) \\[3ex] = \dfrac{1}{2 - \sqrt{3}} \\[5ex] = \dfrac{1}{2 - \sqrt{3}} * \dfrac{2 + \sqrt{3}}{2 + \sqrt{3}} \\[5ex] = \dfrac{2 + \sqrt{3}}{(2)^2 - (\sqrt{3})^2} \\[5ex] = \dfrac{2 + \sqrt{3}}{4 - 3} \\[5ex] = \dfrac{2 + \sqrt{3}}{1} \\[5ex] \cot 15^\circ = 2 + \sqrt{3} \\[3ex]$

$105^\circ$
$105^\circ = 135^\circ - 30^\circ$
Difference Formula

$\sin 105 = \sin(135 - 30) \\[3ex] = \sin135\cos30 - \cos135\sin30 \\[3ex] = \dfrac{\sqrt{2}}{2} * \dfrac{\sqrt{3}}{2} - -\dfrac{\sqrt{2}}{2} * \dfrac{1}{2} \\[5ex] = \dfrac{\sqrt{2 * 3}}{2 * 2} + \dfrac{\sqrt{2} * 1 }{2 * 2} \\[5ex] = \dfrac{\sqrt{6}}{4} + \dfrac{\sqrt{2}}{4} = \dfrac{\sqrt{2}}{4} + \dfrac{\sqrt{6}}{4} \\[5ex] \sin 105^\circ = \dfrac{\sqrt{2} + \sqrt{6}}{4}$

$\cos 105 = \cos(135 - 30) \\[3ex] = \cos135\cos30 + \sin135\sin30 \\[3ex] = -\dfrac{\sqrt{2}}{2} * \dfrac{\sqrt{3}}{2} + \dfrac{\sqrt{2}}{2} * \dfrac{1}{2} \\[5ex] = \dfrac{\sqrt{2} * 1 }{2 * 2} - \dfrac{\sqrt{2 * 3}}{2 * 2} \\[5ex] = \dfrac{\sqrt{2}}{4} - \dfrac{\sqrt{6}}{4} \\[5ex] \cos 105^\circ = \dfrac{\sqrt{2} - \sqrt{6}}{4}$

$\tan 105 = \tan(135 - 30) \\[3ex] = \dfrac{\tan135 - \tan30}{1 + \tan135\tan30} \\[5ex] = \left(-1 - \dfrac{\sqrt{3}}{3}\right) \div \left(1 + -1 * \dfrac{\sqrt{3}}{3}\right) \\[5ex] = \left(-\dfrac{3}{3} - \dfrac{\sqrt{3}}{3}\right) \div \left(1 - \dfrac{\sqrt{3}}{3}\right) \\[5ex] = \left(\dfrac{-3 - \sqrt{3}}{3}\right) \div \left(\dfrac{3}{3} - \dfrac{\sqrt{3}}{3}\right) \\[5ex] = \dfrac{-3 - \sqrt{3}}{3} \div \dfrac{3 - \sqrt{3}}{3} \\[5ex] = \dfrac{-3 - \sqrt{3}}{3} * \dfrac{3}{3 - \sqrt{3}} \\[5ex] = \dfrac{-3 - \sqrt{3}}{3 - \sqrt{3}} \\[5ex] = \dfrac{-3 - \sqrt{3}}{3 - \sqrt{3}} * \dfrac{3 + \sqrt{3}}{3 + \sqrt{3}} \\[5ex] = \dfrac{(-3 - \sqrt{3})(3 + \sqrt{3})}{(3 - \sqrt{3})(3 + \sqrt{3})} \\[5ex] = \dfrac{-9 - 3\sqrt{3} - 3\sqrt{3} - (\sqrt{3})^2}{3^2 - (\sqrt{3})^2} \\[5ex] = \dfrac{-9 - 6\sqrt{3} - 3}{9 - 3} \\[5ex] = \dfrac{-12 - 6\sqrt{3}}{6} = \dfrac{6(-2 - \sqrt{3})}{6} \\[5ex] \tan 105^\circ = -2 - \sqrt{3}$

$\csc 105 = \dfrac{1}{\sin 105} = 1 \div \sin 105 \\[5ex] = 1 \div \dfrac{\sqrt{6} + \sqrt{2}}{4} \\[5ex] = 1 * \dfrac{4}{\sqrt{6} + \sqrt{2}} \\[5ex] = \dfrac{4}{\sqrt{6} + \sqrt{2}} * \dfrac{\sqrt{6} - \sqrt{2}}{\sqrt{6} - \sqrt{2}} \\[5ex] = \dfrac{4(\sqrt{6} - \sqrt{2})}{(\sqrt{6})^2 - (\sqrt{2})^2} \\[5ex] = \dfrac{4(\sqrt{6} - \sqrt{2})}{6 - 2} \\[5ex] = \dfrac{4(\sqrt{6} - \sqrt{2})}{4} \\[5ex] \csc 105^\circ = \sqrt{6} - \sqrt{2}$

$\sec 105 = \dfrac{1}{\cos 105} = 1 \div \cos 105 \\[5ex] = 1 \div \dfrac{\sqrt{2} - \sqrt{6}}{4} \\[5ex] = 1 * \dfrac{4}{\sqrt{2} - \sqrt{6}} \\[5ex] = \dfrac{4}{\sqrt{2} - \sqrt{6}} * \dfrac{\sqrt{2} + \sqrt{6}}{\sqrt{2} + \sqrt{6}} \\[5ex] = \dfrac{4(\sqrt{2} + \sqrt{6})}{(\sqrt{2})^2 - (\sqrt{6})^2} \\[5ex] = \dfrac{4(\sqrt{2} + \sqrt{6})}{2 - 6} \\[5ex] = \dfrac{4(\sqrt{2} + \sqrt{6})}{-4} \\[5ex] = -1(\sqrt{2} + \sqrt{6}) \\[5ex] \sec 105^\circ = -\sqrt{2} - \sqrt{6}$

$\cot 105 = \dfrac{1}{\tan 105} = 1 \div \tan 105 \\[5ex] = 1 \div (-2 - \sqrt{3}) \\[3ex] = \dfrac{1}{-2 - \sqrt{3}} \\[5ex] = \dfrac{1}{-2 - \sqrt{3}} * \dfrac{-2 + \sqrt{3}}{-2 + \sqrt{3}} \\[5ex] = \dfrac{-2 + \sqrt{3}}{(-2)^2 - (\sqrt{3})^2} \\[5ex] = \dfrac{-2 + \sqrt{3}}{4 - 3} \\[5ex] = \dfrac{-2 + \sqrt{3}}{1} \\[5ex] = -2 + \sqrt{3} \\[3ex] \cot 105^\circ = \sqrt{3} - 2$

$165^\circ$
$165^\circ = 135^\circ + 30^\circ$
Sum Formula

$\sin 165 = \sin(135 + 30) \\[3ex] = \sin135\cos30 + \cos135\sin30 \\[3ex] = \dfrac{\sqrt{2}}{2} * \dfrac{\sqrt{3}}{2} + -\dfrac{\sqrt{2}}{2} * \dfrac{1}{2} \\[5ex] = \dfrac{\sqrt{2 * 3}}{2 * 2} - \dfrac{\sqrt{2} * 1 }{2 * 2} \\[5ex] = \dfrac{\sqrt{6}}{4} - \dfrac{\sqrt{2}}{4} \\[5ex] \sin 165^\circ = \dfrac{\sqrt{6} - \sqrt{2}}{4}$

$\cos 165 = \cos(135 + 30) \\[3ex] = \cos135\cos30 - \sin135\sin30 \\[3ex] = -\dfrac{\sqrt{2}}{2} * \dfrac{\sqrt{3}}{2} - \dfrac{\sqrt{2}}{2} * \dfrac{1}{2} \\[5ex] = -\dfrac{\sqrt{2 * 3}}{2 * 2} - \dfrac{\sqrt{2} * 1 }{2 * 2} \\[5ex] = -\dfrac{\sqrt{6}}{4} - \dfrac{\sqrt{2}}{4} \\[5ex] \cos 165^\circ = \dfrac{-\sqrt{6} - \sqrt{2}}{4}$

$\tan 165 = \tan(135 + 30) \\[3ex] = \dfrac{\tan135 + \tan30}{1 - \tan135\tan30} \\[5ex] = \left(-1 + \dfrac{\sqrt{3}}{3}\right) \div \left(1 - -1 * \dfrac{\sqrt{3}}{3}\right) \\[5ex] = \left(-\dfrac{3}{3} + \dfrac{\sqrt{3}}{3}\right) \div \left(1 + \dfrac{\sqrt{3}}{3}\right) \\[5ex] = \left(\dfrac{-3 + \sqrt{3}}{3}\right) \div \left(\dfrac{3}{3} + \dfrac{\sqrt{3}}{3}\right) \\[5ex] = \dfrac{-3 + \sqrt{3}}{3} \div \dfrac{3 + \sqrt{3}}{3} \\[5ex] = \dfrac{-3 + \sqrt{3}}{3} * \dfrac{3}{3 + \sqrt{3}} \\[5ex] = \dfrac{-3 + \sqrt{3}}{3 + \sqrt{3}} \\[5ex] = \dfrac{-3 + \sqrt{3}}{3 + \sqrt{3}} * \dfrac{3 - \sqrt{3}}{3 - \sqrt{3}} \\[5ex] = \dfrac{(-3 + \sqrt{3})(3 - \sqrt{3})}{(3 + \sqrt{3})(3 - \sqrt{3})} \\[5ex] = \dfrac{-9 + 3\sqrt{3} + 3\sqrt{3} - (\sqrt{3})^2}{3^2 - (\sqrt{3})^2} \\[5ex] = \dfrac{-9 + 6\sqrt{3} - 3}{9 - 3} \\[5ex] = \dfrac{-12 + 6\sqrt{3}}{6} \\[5ex] = \dfrac{6(-2 + \sqrt{3})}{6} \\[5ex] = -2 + \sqrt{3} \\[3ex] \tan 165^\circ = \sqrt{3} - 2$

$\csc 165 = \dfrac{1}{\sin 165} = 1 \div \sin165 \\[5ex] = 1 \div \dfrac{\sqrt{6} - \sqrt{2}}{4} \\[5ex] = 1 * \dfrac{4}{\sqrt{6} - \sqrt{2}} \\[5ex] = \dfrac{4}{\sqrt{6} - \sqrt{2}} * \dfrac{\sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}} \\[5ex] = \dfrac{4(\sqrt{6} + \sqrt{2})}{(\sqrt{6})^2 - (\sqrt{2})^2} \\[5ex] = \dfrac{4(\sqrt{6} + \sqrt{2})}{6 - 2} \\[5ex] = \dfrac{4(\sqrt{6} + \sqrt{2})}{4} \\[5ex] \csc 165^\circ = \sqrt{6} + \sqrt{2}$

$\sec 165 = \dfrac{1}{\cos 165} = 1 \div \cos165 \\[5ex] = 1 \div \dfrac{-\sqrt{6} - \sqrt{2}}{4} \\[5ex] = 1 * \dfrac{4}{-\sqrt{6} - \sqrt{2}} \\[5ex] = \dfrac{4}{-\sqrt{6} - \sqrt{2}} * \dfrac{-\sqrt{6} + \sqrt{2}}{-\sqrt{6} + \sqrt{2}} \\[5ex] = \dfrac{4(-\sqrt{6} + \sqrt{2})}{(-\sqrt{6})^2 - (\sqrt{2})^2} \\[5ex] = \dfrac{4(-\sqrt{6} + \sqrt{2})}{6 - 2} \\[5ex] = \dfrac{4(-\sqrt{6} + \sqrt{2})}{4} \\[5ex] = -\sqrt{6} + \sqrt{2} \\[3ex] \sec 165^\circ = \sqrt{2} - \sqrt{6}$

$\cot 165 = \dfrac{1}{\tan 165} = 1 \div \tan165 \\[5ex] = 1 \div (-2 + \sqrt{3}) \\[3ex] = \dfrac{1}{-2 + \sqrt{3}} \\[5ex] = \dfrac{1}{-2 + \sqrt{3}} * \dfrac{-2 - \sqrt{3}}{-2 - \sqrt{3}} \\[5ex] = \dfrac{-2 - \sqrt{3}}{(-2)^2 - (\sqrt{3})^2} \\[5ex] = \dfrac{-2 - \sqrt{3}}{4 - 3} \\[5ex] = \dfrac{-2 - \sqrt{3}}{1} \\[5ex] \cot 165^\circ = -2 - \sqrt{3} \\$

$255^\circ$
$255^\circ = 210^\circ + 45^\circ$
Sum Formula

$\sin 255 = \sin(210 + 45) \\[3ex] = \sin210\cos45 + \cos210\sin45 \\[3ex] = -\dfrac{1}{2} * \dfrac{\sqrt{2}}{2} + -\dfrac{\sqrt{3}}{2} * \dfrac{\sqrt{2}}{2} \\[5ex] = -\dfrac{1 * \sqrt{2}}{2 * 2} - \dfrac{\sqrt{3 * 2}}{2 * 2} \\[5ex] = -\dfrac{\sqrt{2}}{4} - \dfrac{\sqrt{6}}{4} \\[5ex] \sin 255^\circ = \dfrac{-\sqrt{2} - \sqrt{6}}{4}$

$\cos 255 = \cos(210 + 45) \\[3ex] = \cos210\cos45 - \sin210\sin45 \\[3ex] = -\dfrac{\sqrt{3}}{2} * \dfrac{\sqrt{2}}{2} - -\dfrac{1}{2} * \dfrac{\sqrt{2}}{2} \\[5ex] = -\dfrac{\sqrt{3 * 2}}{2 * 2} + \dfrac{1 * \sqrt{2}}{2 * 2} \\[5ex] = -\dfrac{\sqrt{6}}{4} + \dfrac{\sqrt{2}}{4} \\[5ex] \cos 255^\circ = \dfrac{\sqrt{2} - \sqrt{6}}{4}$

$\tan 255 = \tan(210 + 45) \\[3ex] = \dfrac{\tan210 + \tan45}{1 - \tan210\tan45} \\[5ex] = \left(\dfrac{\sqrt{3}}{3} + 1\right) \div \left(1 - \dfrac{\sqrt{3}}{3} * 1\right) \\[5ex] = \left(\dfrac{\sqrt{3}}{3} + \dfrac{3}{3}\right) \div \left(1 - \dfrac{\sqrt{3}}{3}\right) \\[5ex] = \left(\dfrac{\sqrt{3} + 3}{3}\right) \div \left(\dfrac{3}{3} - \dfrac{\sqrt{3}}{3}\right) \\[5ex] = \dfrac{\sqrt{3} + 3}{3} \div \dfrac{3 - \sqrt{3}}{3} \\[5ex] = \dfrac{\sqrt{3} + 3}{3} * \dfrac{3}{3 - \sqrt{3}} \\[5ex] = \dfrac{\sqrt{3} + 3}{3 - \sqrt{3}} \\[5ex] = \dfrac{\sqrt{3} + 3}{3 - \sqrt{3}} * \dfrac{3 + \sqrt{3}}{3 + \sqrt{3}} \\[5ex] = \dfrac{3\sqrt{3} + (\sqrt{3})^2 + 9 + 3\sqrt{3}}{3^2 - (\sqrt{3})^2} \\[5ex] = \dfrac{3\sqrt{3} + 3 + 9 + 3\sqrt{3}}{9 - 3} \\[5ex] = \dfrac{12 + 6\sqrt{3}}{6} \\[5ex] = \dfrac{6(2 + \sqrt{3})}{6} \\[5ex] = 2 + \sqrt{3} \\[3ex] \tan 255^\circ = 2 + \sqrt{3}$

$\csc 255 = \dfrac{1}{\sin 255} = 1 \div \sin255 \\[5ex] = 1 \div \dfrac{-\sqrt{2} - \sqrt{6}}{4} \\[5ex] = 1 * \dfrac{4}{-\sqrt{2} - \sqrt{6}} \\[5ex] = \dfrac{4}{-\sqrt{2} - \sqrt{6}} * \dfrac{-\sqrt{2} + \sqrt{6}}{-\sqrt{2} + \sqrt{6}} \\[5ex] = \dfrac{4(-\sqrt{2} + \sqrt{6})}{(-\sqrt{2})^2 - (\sqrt{6})^2} \\[5ex] = \dfrac{4(-\sqrt{2} + \sqrt{6})}{2 - 6} \\[5ex] = \dfrac{4(-\sqrt{2} + \sqrt{6})}{-4} \\[5ex] = -1(-\sqrt{2} + \sqrt{6}) \\[3ex] = \sqrt{2} - \sqrt{6} \\[3ex] \csc 255^\circ = \sqrt{2} - \sqrt{6}$

$\sec 255 = \dfrac{1}{\cos 255} = 1 \div \cos255 \\[5ex] = 1 \div \dfrac{\sqrt{2} - \sqrt{6}}{4} \\[5ex] = 1 * \dfrac{4}{\sqrt{2} - \sqrt{6}} \\[5ex] = \dfrac{4}{\sqrt{2} - \sqrt{6}} * \dfrac{\sqrt{2} + \sqrt{6}}{\sqrt{2} + \sqrt{6}} \\[5ex] = \dfrac{4(\sqrt{2} + \sqrt{6})}{(\sqrt{2})^2 - (\sqrt{6})^2} \\[5ex] = \dfrac{4(\sqrt{2} + \sqrt{6})}{2 - 6} \\[5ex] = \dfrac{4(\sqrt{2} + \sqrt{6})}{-4} \\[5ex] = -1(\sqrt{2} + \sqrt{6}) = -\sqrt{2} - \sqrt{6} \\[3ex] \sec 255^\circ = -\sqrt{2} - \sqrt{6}$

$\cot 255 = \dfrac{1}{\tan 255} = 1 \div \tan255 \\[5ex] = 1 \div (2 + \sqrt{3}) \\[3ex] = \dfrac{1}{2 + \sqrt{3}} \\[5ex] = \dfrac{1}{2 + \sqrt{3}} * \dfrac{2 - \sqrt{3}}{2 - \sqrt{3}} \\[5ex] = \dfrac{2 - \sqrt{3}}{(2)^2 - (\sqrt{3})^2} \\[5ex] = \dfrac{2 - \sqrt{3}}{4 - 3} \\[5ex] = \dfrac{2 - \sqrt{3}}{1} \\[5ex] \cot 255^\circ = 2 - \sqrt{3}$

$195^\circ$
$195^\circ = 225^\circ - 30^\circ$
Difference Formula

$\sin 195 = \sin(225 - 30) \\[3ex] = \sin225\cos30 - \cos225\sin30 \\[3ex] = -\dfrac{\sqrt{2}}{2} * \dfrac{\sqrt{3}}{2} - -\dfrac{\sqrt{2}}{2} * \dfrac{1}{2} \\[5ex] = -\dfrac{\sqrt{2 * 3}}{2 * 2} + \dfrac{\sqrt{2} * 1 }{2 * 2} \\[5ex] = - \dfrac{\sqrt{6}}{4} + \dfrac{\sqrt{2}}{4} \\[5ex] = \dfrac{\sqrt{2}}{4} - \dfrac{\sqrt{6}}{4} \\[5ex] \sin 195^\circ = \dfrac{\sqrt{2} - \sqrt{6}}{4}$

$\cos 195 = \cos(225 - 30) \\[3ex] = \cos225\cos30 + \sin225\sin30 \\[3ex] = -\dfrac{\sqrt{2}}{2} * \dfrac{\sqrt{3}}{2} + -\dfrac{\sqrt{2}}{2} * \dfrac{1}{2} \\[5ex] = -\dfrac{\sqrt{2 * 3}}{2 * 2} - \dfrac{\sqrt{2} * 1 }{2 * 2} \\[5ex] = -\dfrac{\sqrt{6}}{4} - \dfrac{\sqrt{2}}{4} \\[5ex] \cos 195^\circ = \dfrac{-\sqrt{6} - \sqrt{2}}{4}$

$\tan 195 = \tan(225 - 30) \\[3ex] = \dfrac{\tan225 - \tan30}{1 + \tan225\tan30} \\[5ex] = \left(1 - \dfrac{\sqrt{3}}{3}\right) \div \left(1 + 1 * \dfrac{\sqrt{3}}{3}\right) \\[5ex] = \left(\dfrac{3}{3} - \dfrac{\sqrt{3}}{3}\right) \div \left(1 + \dfrac{\sqrt{3}}{3}\right) \\[5ex] = \left(\dfrac{3 - \sqrt{3}}{3}\right) \div \left(\dfrac{3}{3} + \dfrac{\sqrt{3}}{3}\right) \\[5ex] = \dfrac{3 - \sqrt{3}}{3} \div \dfrac{3 + \sqrt{3}}{3} \\[5ex] = \dfrac{3 - \sqrt{3}}{3} * \dfrac{3}{3 + \sqrt{3}} \\[5ex] = \dfrac{3 - \sqrt{3}}{3 + \sqrt{3}} \\[5ex] = \dfrac{3 - \sqrt{3}}{3 + \sqrt{3}} * \dfrac{3 - \sqrt{3}}{3 - \sqrt{3}} \\[5ex] = \dfrac{(3 - \sqrt{3})(3 - \sqrt{3})}{(3 + \sqrt{3})(3 - \sqrt{3})} \\[5ex] = \dfrac{9 - 3\sqrt{3} - 3\sqrt{3} + (\sqrt{3})^2}{3^2 - (\sqrt{3})^2} \\[5ex] = \dfrac{9 - 6\sqrt{3} + 3}{9 - 3} \\[5ex] = \dfrac{12 - 6\sqrt{3}}{6} \\[5ex] = \dfrac{6(2 - \sqrt{3})}{6} \\[5ex] \tan 195^\circ = 2 - \sqrt{3}$

$\csc 195 = \dfrac{1}{\sin 195} = 1 \div \sin195 \\[5ex] = 1 \div \dfrac{\sqrt{2} - \sqrt{6}}{4} \\[5ex] = 1 * \dfrac{4}{\sqrt{2} - \sqrt{6}} \\[5ex] = \dfrac{4}{\sqrt{2} - \sqrt{6}} * \dfrac{\sqrt{2} + \sqrt{6}}{\sqrt{2} + \sqrt{6}} \\[5ex] = \dfrac{4(\sqrt{2} + \sqrt{6})}{(\sqrt{2})^2 - (\sqrt{6})^2} \\[5ex] = \dfrac{4(\sqrt{2} + \sqrt{6})}{2 - 6} \\[5ex] = \dfrac{4(\sqrt{2} + \sqrt{6})}{-4} \\[5ex] = -1(\sqrt{2} + \sqrt{6}) \\[5ex] \csc 195^\circ = -\sqrt{2} - \sqrt{6}$

$\sec 195 = \dfrac{1}{\cos 195} = 1 \div \cos195 \\[5ex] = 1 \div \dfrac{-\sqrt{6} - \sqrt{2}}{4} \\[5ex] = 1 * \dfrac{4}{-\sqrt{6} - \sqrt{2}} \\[5ex] = \dfrac{4}{-\sqrt{6} - \sqrt{2}} * \dfrac{-\sqrt{6} + \sqrt{2}}{-\sqrt{6} + \sqrt{2}} \\[5ex] = \dfrac{4(-\sqrt{6} + \sqrt{2})}{(-\sqrt{6})^2 - (\sqrt{2})^2} \\[5ex] = \dfrac{4(-\sqrt{6} + \sqrt{2})}{6 - 2} \\[5ex] = \dfrac{4(-\sqrt{6} + \sqrt{2})}{4} \\[5ex] = -\sqrt{6} + \sqrt{2} \\[3ex] \sec 195^\circ = \sqrt{2} - \sqrt{6}$

$\cot 195 = \dfrac{1}{\tan 195} = 1 \div \tan195 \\[5ex] = 1 \div (2 - \sqrt{3}) \\[3ex] = \dfrac{1}{2 - \sqrt{3}} \\[5ex] = \dfrac{1}{2 - \sqrt{3}} * \dfrac{2 + \sqrt{3}}{2 + \sqrt{3}} \\[5ex] = \dfrac{2 + \sqrt{3}}{(2)^2 - (\sqrt{3})^2} \\[5ex] = \dfrac{2 + \sqrt{3}}{4 - 3} = \dfrac{2 + \sqrt{3}}{1} \\[5ex] \cot 195^\circ = 2 + \sqrt{3}$

$285^\circ$
$285^\circ = 330^\circ - 45^\circ$
Difference Formula

$\sin 285 = \sin(330 - 45) \\[3ex] = \sin330\cos45 - \cos330\sin45 \\[3ex] = -\dfrac{1}{2} * \dfrac{\sqrt{2}}{2} - \dfrac{\sqrt{3}}{2} * \dfrac{\sqrt{2}}{2} \\[5ex] = \dfrac{-1 * \sqrt{2}}{2 * 2} - \dfrac{\sqrt{3} * \sqrt{2}}{2 * 2} \\[5ex] = \dfrac{-\sqrt{2}}{4} - \dfrac{\sqrt{6}}{4} \\[5ex] \sin 285^\circ = \dfrac{-\sqrt{2} - \sqrt{6}}{4}$

$\cos 285 = \cos(330 - 45) \\[3ex] = \cos330\cos45 + \sin330\sin45 \\[3ex] = \dfrac{\sqrt{3}}{2} * \dfrac{\sqrt{2}}{2} + -\dfrac{1}{2} * \dfrac{\sqrt{2}}{2} \\[5ex] = \dfrac{\sqrt{3} * \sqrt{2}}{2 * 2} - \dfrac{1 * \sqrt{2}}{2 * 2} \\[5ex] = \dfrac{\sqrt{6}}{4} - \dfrac{\sqrt{2}}{4} \\[5ex] \cos 285^\circ = \dfrac{\sqrt{6} - \sqrt{2}}{4}$

$\tan 285 = \tan(330 - 45) \\[3ex] = \dfrac{\tan330 - \tan45}{1 + \tan330\tan45} \\[5ex] = \left(-\dfrac{\sqrt{3}}{3} - 1\right) \div \left(1 + -\dfrac{\sqrt{3}}{3} * 1\right) \\[5ex] = \left(-\dfrac{\sqrt{3}}{3} -\dfrac{3}{3}\right) \div \left(1 - \dfrac{\sqrt{3}}{3}\right) \\[5ex] = \left(\dfrac{-\sqrt{3} - 3}{3}\right) \div \left(\dfrac{3}{3} - \dfrac{\sqrt{3}}{3}\right) \\[5ex] = \dfrac{-\sqrt{3} - 3}{3} \div \dfrac{3 - \sqrt{3}}{3} \\[5ex] = \dfrac{-\sqrt{3} - 3}{3} * \dfrac{3}{3 - \sqrt{3}} \\[5ex] = \dfrac{-\sqrt{3} - 3}{3 - \sqrt{3}} \\[5ex] = \dfrac{-\sqrt{3} - 3}{3 - \sqrt{3}} * \dfrac{3 + \sqrt{3}}{3 + \sqrt{3}} \\[5ex] = \dfrac{(-\sqrt{3} - 3)(3 + \sqrt{3})}{(3 - \sqrt{3})(3 + \sqrt{3})} \\[5ex] = \dfrac{-3\sqrt{3} - (\sqrt{3})^2 - 9 - 3\sqrt{3}}{3^2 - (\sqrt{3})^2} \\[5ex] = \dfrac{- 3 - 9 - 6\sqrt{3} }{9 - 3} \\[5ex] = \dfrac{-12 - 6\sqrt{3}}{6} = \dfrac{6(-2 - \sqrt{3})}{6} \\[5ex] \tan 285^\circ = -2 - \sqrt{3}$

$\csc 285 = \dfrac{1}{\sin 285} = 1 \div \sin 285 \\[5ex] = 1 \div \dfrac{-\sqrt{2} - \sqrt{6}}{4} \\[5ex] = 1 * \dfrac{4}{-\sqrt{2} - \sqrt{6}} \\[5ex] = \dfrac{4}{-\sqrt{2} - \sqrt{6}} * \dfrac{-\sqrt{2} + \sqrt{6}}{-\sqrt{2} + \sqrt{6}} \\[5ex] = \dfrac{4(-\sqrt{2} + \sqrt{6})}{(-\sqrt{2})^2 - (\sqrt{6})^2} \\[5ex] = \dfrac{4(-\sqrt{2} + \sqrt{6})}{2 - 6} \\[5ex] = \dfrac{4(-\sqrt{2} + \sqrt{6})}{-4} \\[5ex] = -1(-\sqrt{2} + \sqrt{6}) \\[3ex] = \sqrt{2} - \sqrt{6} \\[3ex] \csc 285^\circ = \sqrt{2} - \sqrt{6}$

$\sec 285 = \dfrac{1}{\cos 285} = 1 \div \cos 285 \\[5ex] = 1 \div \dfrac{\sqrt{6} - \sqrt{2}}{4} \\[5ex] = 1 * \dfrac{4}{\sqrt{6} - \sqrt{2}} \\[5ex] = \dfrac{4}{\sqrt{6} - \sqrt{2}} * \dfrac{\sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}} \\[5ex] = \dfrac{4(\sqrt{6} + \sqrt{2})}{(\sqrt{6})^2 - (\sqrt{2})^2} \\[5ex] = \dfrac{4(\sqrt{6} + \sqrt{2})}{6 - 2} \\[5ex] = \dfrac{4(\sqrt{6} + \sqrt{2}}{4} \\[5ex] \sec 285^\circ = \sqrt{6} + \sqrt{2}$

$\cot 285 = \dfrac{1}{\tan 285} = 1 \div \tan 285 \\[5ex] = 1 \div (-2 - \sqrt{3}) \\[3ex] = \dfrac{1}{-2 - \sqrt{3}} \\[5ex] = \dfrac{1}{-2 - \sqrt{3}} * \dfrac{-2 + \sqrt{3}}{-2 + \sqrt{3}} \\[5ex] = \dfrac{-2 + \sqrt{3}}{(-2)^2 - (\sqrt{3})^2} \\[5ex] = \dfrac{-2 + \sqrt{3}}{4 - 3} \\[5ex] = \dfrac{-2 + \sqrt{3}}{1} \\[5ex] = -2 + \sqrt{3} \\[3ex] \cot 285^\circ = \sqrt{3} - 2$

$345^\circ$
$345^\circ = 300^\circ + 45^\circ$
Sum Formula

$\sin 345 = \sin(300 + 45) \\[3ex] = \sin300\cos45 + \cos300\sin45 \\[3ex] = -\dfrac{\sqrt{3}}{2} * \dfrac{\sqrt{2}}{2} + \dfrac{1}{2} * \dfrac{\sqrt{2}}{2} \\[5ex] = -\dfrac{\sqrt{3 * 2}}{2 * 2} + \dfrac{1 * \sqrt{2}}{2 * 2} \\[5ex] = -\dfrac{\sqrt{6}}{4} + \dfrac{\sqrt{2}}{4} \\[5ex] = \dfrac{\sqrt{2}}{4} - \dfrac{\sqrt{6}}{4} \\[5ex] \sin 345^\circ = \dfrac{\sqrt{2} - \sqrt{6}}{4}$

$\cos 345 = \cos(300 + 45) \\[3ex] = \cos300\cos45 - \sin300\sin45 \\[3ex] = \dfrac{1}{2} * \dfrac{\sqrt{2}}{2} - -\dfrac{\sqrt{3}}{2} * \dfrac{\sqrt{2}}{2} \\[5ex] = \dfrac{1 * \sqrt{2}}{2 * 2} + \dfrac{\sqrt{3 * 2}}{2 * 2} \\[5ex] = \dfrac{\sqrt{2}}{4} + \dfrac{\sqrt{6}}{4} \\[5ex] \cos 345^\circ = \dfrac{\sqrt{2} + \sqrt{6}}{4}$

$\tan 345 = \tan(300 + 45) \\[3ex] = \dfrac{\tan300 + \tan45}{1 - \tan300\tan45} \\[5ex] = \dfrac{-\sqrt{3} + 1}{1 - (-\sqrt{3})(1)} \\[5ex] = \dfrac{1 - \sqrt{3}}{1 - -\sqrt{3}} \\[5ex] = \dfrac{1 - \sqrt{3}}{1 + \sqrt{3}} \\[5ex] = \dfrac{1 - \sqrt{3}}{1 + \sqrt{3}} * \dfrac{1 - \sqrt{3}}{1 - \sqrt{3}} \\[5ex] = \dfrac{(1 - \sqrt{3})(1 - \sqrt{3})}{(1 + \sqrt{3})(1 - \sqrt{3})} \\[5ex] = \dfrac{1 - \sqrt{3} - \sqrt{3} + (\sqrt{3})^2}{1^2 - (\sqrt{3})^2} \\[5ex] = \dfrac{1 - 2\sqrt{3} + 3}{1 - 3} \\[5ex] = \dfrac{4 - 2\sqrt{3}}{-2} \\[5ex] = \dfrac{2(2 - \sqrt{3})}{-2} \\[5ex] = -(2 - \sqrt{3}) = -2 + \sqrt{3} \\[3ex] \tan 345^\circ = \sqrt{3} - 2$

$\csc 345 = \dfrac{1}{\sin345} = 1 \div \sin345 \\[5ex] = 1 \div \dfrac{\sqrt{2} - \sqrt{6}}{4} \\[5ex] = 1 * \dfrac{4}{\sqrt{2} - \sqrt{6}} \\[5ex] = \dfrac{4}{\sqrt{2} - \sqrt{6}} * \dfrac{\sqrt{2} + \sqrt{6}}{\sqrt{2} + \sqrt{6}} \\[5ex] = \dfrac{4(\sqrt{2} + \sqrt{6})}{(\sqrt{2})^2 - (\sqrt{6})^2} \\[5ex] = \dfrac{4(\sqrt{2} + \sqrt{6})}{2 - 6} \\[5ex] = \dfrac{4(\sqrt{2} + \sqrt{6})}{-4} \\[5ex] = -(\sqrt{2} + \sqrt{6}) = -\sqrt{2} - \sqrt{6} \\[3ex] \csc 345^\circ = -\sqrt{2} - \sqrt{6}$

$\sec 345 = \dfrac{1}{\cos 345} = 1 \div \cos345 \\[5ex] = 1 \div \dfrac{\sqrt{2} + \sqrt{6}}{4} \\[5ex] = 1 * \dfrac{4}{\sqrt{2} + \sqrt{6}} \\[5ex] = \dfrac{4}{\sqrt{2} + \sqrt{6}} * \dfrac{\sqrt{2} - \sqrt{6}}{\sqrt{2} - \sqrt{6}} \\[5ex] = \dfrac{4(\sqrt{2} - \sqrt{6})}{(\sqrt{2})^2 - (\sqrt{6})^2} \\[5ex] = \dfrac{4(\sqrt{2} - \sqrt{6})}{2 - 6} \\[5ex] = \dfrac{4(\sqrt{2} - \sqrt{6})}{-4} \\[5ex] = -(\sqrt{2} - \sqrt{6}) = -\sqrt{2} + \sqrt{6} \\[3ex] \sec 345^\circ = \sqrt{6} - \sqrt{2}$

$\cot 345 = \dfrac{1}{\tan 345} = 1 \div \tan345 \\[5ex] = 1 \div (\sqrt{3} - 2) \\[3ex] = \dfrac{1}{\sqrt{3} - 2} \\[5ex] = \dfrac{1}{\sqrt{3} - 2} * \dfrac{\sqrt{3} + 2}{\sqrt{3} + 2} \\[5ex] = \dfrac{\sqrt{3} + 2}{(\sqrt{3})^2 - (2)^2} \\[5ex] = \dfrac{\sqrt{3} + 2}{3 - 4} \\[5ex] = \dfrac{\sqrt{3} + 2}{-1} \\[5ex] = -(\sqrt{3} + 2) = -\sqrt{3} - 2 \\[3ex] \cot 345^\circ = -2 - \sqrt{3} \\[3ex]$