9.2E: Sum and Difference Identities (Exercises)
For the following exercises, find the exact value.
11. \(\tan \left(\frac{7 \pi}{12}\right)\)
12. \(\cos \left(\frac{25 \pi}{12}\right)\)
13. \(\sin \left(70^{\circ}\right) \cos \left(25^{\circ}\right)-\cos \left(70^{\circ}\right) \sin \left(25^{\circ}\right)\)
14. \(\cos \left(83^{\circ}\right) \cos \left(23^{\circ}\right)+\sin \left(83^{\circ}\right) \sin \left(23^{\circ}\right)\)
For the following exercises, prove the identity.
15. \(\cos (4 x)-\cos (3 x) \cos x=\sin ^{2} x-4 \cos ^{2} x \sin ^{2} x\)
16. \(\cos (3 x)-\cos ^{3} x=-\cos x \sin ^{2} x-\sin x \sin (2 x)\)
For the following exercise, simplify the expression.
17. \(\frac{\tan \left(\frac{1}{2} x\right)+\tan \left(\frac{1}{8} x\right)}{1-\tan \left(\frac{1}{8} x\right) \tan \left(\frac{1}{2} x\right)}\)
For the following exercises, find the exact value.
18. \(\cos \left(\sin ^{-1}(0)-\cos ^{-1}\left(\frac{1}{2}\right)\right)\)
19. \(\tan \left(\sin ^{-1}(0)+\sin ^{-1}\left(\frac{1}{2}\right)\right)\)