Trigonometry test 2

1. Find the values of following:

   1. \(\cos{75^\circ}\)
   2. \(\tan{\frac{5\pi}{12}}\)
   3. \(\sin{120^\circ}\)
   4. \(\cot{\frac{3\pi}{4}}\)
   5. \(\sec{225^\circ}\)

2. Simplify the following:

   1. \(\cos{(\frac{3\pi}{2}+x)}\)
   2. \(\tan{(\pi-x)}\)
   3. \(\sec{(\frac{\pi}{2}+x)}\)
   4. \(\sin{(2\pi-x)}\)

3. Prove the following:

   1. \(2(\sin{45^\circ}\cos{15^\circ}-\cos{45^\circ}\sin{15^\circ})=1\)
   2. \(\sin{15^\circ}\cos{15^\circ}=\frac{1}{4}\) [Hint: \(\sin{2x}\)]
   3. \(\sqrt{2}\sin{(\frac{\pi}{4}+x)}=\sin{x}+\cos{x}\)
   4. \(\frac{\tan{(n+1)x-\tan{nx}}}{1+\tan{(n+1)x}\tan{nx}}=\tan{x}\)
   5. \(\cot{x}-\tan{x}=2\cot{2x}\)
   6. \(\frac{\sin{7x}-\sin{3x}}{\sin{x}\cos{x}}=2\cos{5x}\)
   7. \(\frac{\sin{6x}-\sin{2x}}{\cos{5x}+\cos{3x}}=2\sin{x}\)
   8. \(2\sin{(\frac{3\pi}{4}+x)}\cos{(\frac{\pi}{4}-x)}=\cos{2x}\)