(Solved): only have to do one part from each question 1. Find the exact value of each of the following. (a) co ...

1. Find the exact value of each of the following. (a) $\cos \left(-\frac{\pi }{12}\right)$ (b) $\sin \frac{3}{2}\pi$ (c) $\frac{\tan 67^{\circ }-\tan 22^{\circ }}{1+\tan 67^{\circ }\tan 22^{\circ }}$ (d) ${\left(\sin \frac{\pi }{8}+\cos \frac{\pi }{8}\right)}^2$ (e) $\sin \frac{13}{46}\pi \cos \frac{5}{36}\pi +\cos \frac{13}{36}\pi \sin \frac{5}{5}\pi$ 2. If $\sin x=\frac{12}{13},x$ in the interval $\left[0,\frac{\pi }{2}\right]$ and $\cos y=\frac{4}{5},y$ in the interval $\left[-\frac{\pi }{2},0\right]$, find the value of $\sin [2(x-y)]$. 3. Find the value of $\tan 2x,\frac{\pi }{2}<x<\pi$, given $\sec x=-\frac{5}{4}$ 4. If $\sqrt{2}\cos x-1=\frac{1+\sqrt{3}}{2}$ and $\sqrt{2}\cos x+1=\frac{1-\sqrt{3}}{2}$, find the value of $\cos 4x$. 5. Prove. (a) $\frac{1+\sin x+\cos x}{1+\sin x-\cos x}=\cot \frac{x}{2}$ (b) $-{\sin }^{2}x-{\sin }^{2}y+1=\cos (x+y)\cos (x-y)$ (c) $\frac{\sin (x-\pi )}{\cos (\pi +x)}-\frac{\cos \left(\frac{\pi }{2}-x\right)}{\sin (-\pi -x)}=\frac{\sin x-\cos x}{\cos x}$ 6. Solve for $x$ in the given interval. (a) ${\tan }^2\left(2x-\frac{\pi }{12}\right)=3,\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$ (b) $\cos 2x-{\cos }^{2}x-2\sin x+3=0,[0,2\pi ]$ 7. Develop a formula for $\tan (a+b-c)$ in terms of $\tan a,\tan b$, and $\tan c$.