Show that $${{\sin \theta } \over {\cos 3\theta }} + {{\sin 3\theta } \over {\cos 9\theta }} + {{\sin 9\theta } \over {\cos 27\theta }} = {1 \over 2}(\tan 27\theta - \tan \theta )$$

Prov that the equation $$\cos 2x + a\sin x = 2a - 7$$ possesses a solution if 2 $$\le$$ a $$\le$$ 6.

Find the values of x, ($$-$$ $$\pi$$, < x < $$\pi$$, x $$\ne$$ 0) satisfying the equation, $${8^{1 + |\cos x| + |{{\cos }^2}x| + ......\infty }} = {4^3}$$.