1. $2{\cos ^4}x + 2{\cos ^4}\left( {\frac{\pi }{4} - x} \right) = 2 - \sin 2x$
2. $2\cos x + \sin x\left( {\tan \frac{x}{2} + \cot x} \right) = 2$
3. $2\cos 2x\sin x + \cos x\left( {1 + \tan x\tan \frac{x}{2}} \right) = 2{\cos ^2}x\left( {\frac{\pi }{4} - \frac{{3x}}{2}} \right) + \frac{{\sqrt 3 }}{2}$
4. $\cos x + {\sin ^2}x\left( {1 + \cot x\cot \frac{x}{2}} \right) = 5$
5. ${\sin ^4}\left( {x + \frac{\pi }{4}} \right) + {\sin ^4}x = \frac{1}{4}$