Find the range of values of $$\,t$$ for which $$$2\,\sin \,t = {{1 - 2x + 5{x^2}} \over {3{x^2} - 2x - 1}},\,\,\,\,\,t\, \in \,\left[ { - {\pi \over 2},\,{\pi \over 2}} \right].$$$

In any triangle $$ABC,$$ prove that $$$\cot {A \over 2} + \cot {B \over 2} + \cot {C \over 2} = \cot {A \over 2}\cot {B \over 2}\cot {C \over 2}.$$$

Prove that $$\tan \,\alpha + 2\tan 2\alpha + 4\tan 4\alpha + 8\cot 8\alpha = \cot \alpha $$

Prove that the values of the function $${{\sin x\cos 3x} \over {\sin 3x\cos x}}$$ do not lie between $${1 \over 3}$$ and 3 for any real $$x.$$