Double-angle identities

Lessons

1. Express each of the following in terms of a single trigonometric function:
   
   1. $14 \sin 6x \cos 6x$
      
   2. $\cos {9 \over 2} x \sin {9 \over 2 } x$
      
   3. $\cos^25A - \sin^25A$
      
   4. $3 - 6 \cos^2 8x$
      
   5. $\frac{6 \tan 5x}{\tan^25x - 1}$
      
2. Prove identities
   1. $\frac{\sin 6x}{ 1 + \cos 6x} =\tan 3x$
      
   2. $\cot x - \tan x= \frac{4 \cos^2x - 2}{\sin 2x}$
      
   3. $\frac{\sin 4x - \sin 2x}{\cos 4x + \cos 2x} = \tan x$
      
   4. $\frac{\cot x - \cos x}{ 1 - \sin x} = \frac{\sin2x}{1 - \cos2x}$
      
3. Given $\tan A = -2,$ where ${3 \pi \over 2 } \leq A \leq 2\pi,$
   find the exact value of:
   i)
   $\sin 2A$
   ii)
   $\sec 2A$
   iii) $\tan 2A$