elementary functions

Exercise

1.: Show that the following formulas are true for arbitrary angle $\alpha, \beta$ ．
(a): $\sin{(\alpha \pm \beta)} = \sin{\alpha}\cos{\beta} \pm \cos{\alpha}\sin{\beta}$ (Addition formula)
(b): $\cos{(\alpha \pm \beta)} = \cos{\alpha}\cos{\beta} \mp \sin{\alpha}\sin{\beta}$
(c): $\displaystyle{\cos^{2}{\frac{\alpha}{2}} = \frac{1 + \cos{\alpha}}{2}}$ (Half angle formula)
(d): $\displaystyle{\sin{\alpha}\cos{\beta} = \frac{1}{2}\left(\sin{(\alpha + \beta)} + \sin{(\alpha - \beta)}\right)}$ (Product to sum)
(e): $\displaystyle{\sin{\alpha} + \sin{\beta} = 2 \sin{\frac{\alpha + \beta}{2}} \cos{\frac{\alpha - \beta}{2}}}$ (Sum to product)
2.: Find the value of followings:．
(a): $\displaystyle{\cos{\frac{5\pi}{4}}}$
(b): $\displaystyle{\sin{\frac{7\pi}{12}}}$
(c): $\displaystyle{\cos{\frac{\pi}{8}}}$
3.: Find the value of followings:
(a): $\displaystyle{\sin^{-1}{(\frac{-1}{2})}}$
(b): $\displaystyle{\cos^{-1}{(-1)}}$
(c): $\displaystyle{\tan^{-1}{(-1)}}$
(d): $\displaystyle{\tan^{-1}{\sqrt{3}}}$
4.: For all ，show that $\displaystyle{\sin^{-1}{x} + \cos^{-1}{x} = \frac{\pi}{2}}$ holds．
5.: Derive the following formulae:．
(a): $\displaystyle{\sin^{-1}{(-x)} = - \sin^{-1}{x}}$
(b): $\displaystyle{\cos^{-1}{(-x)} = \pi - \cos^{-1}{x}}$