Question

Identity transformations of Trigonometric Expressions.
prove the following identities.
$cos4\alpha -sin4\alpha cot2\alpha =cos2\alpha -2co{s}^2\alpha .$

Answer 1

For proving the identity,

$\cos 4\alpha -\sin 4\alpha \cos 2\alpha =\cos 2\alpha -2{\cos }^2\alpha$

First evaluate the L.H.S,

$\cos 4\alpha -\sin 4\alpha \cot 2\alpha$

$=\cos 4\alpha -2\left(\sin 2\alpha \cos 2\alpha \right)\cot 2\alpha$ [Using the fact, $\sin 4\alpha =2\sin 2\alpha \cos 2\alpha$]

$=\cos 4\alpha -2\left(\sin 2\alpha \cos 2\alpha \right)\frac{\cos 2\alpha }{\sin 2\alpha }$ [Using the fact, $\cot 2\alpha =\frac{\cos 2\alpha }{\sin 2\alpha }$]

$=\cos 4\alpha -2{\cos }^{2}2\alpha$

$=2{\cos }^{2}2\alpha -1-2{\cos }^{2}2\alpha$ [using the fact, $\cos 4\alpha =2{\cos }^{2}2\alpha -1$]

$=-1$

Now, let's evaluate the R.H.S,

$=\cos 2\alpha -2{\cos }^2\alpha$

$=2{\cos }^2\alpha -1-2{\cos }^2\alpha$ [using the fact, $\cos 2\alpha =2{\cos }^2\alpha -1$]

$=-1$

Since, L.H.S$=$R.H.S, we can say

$\cos 4\alpha -\sin 4\alpha \cot 2\alpha =\cos 2\alpha -2{\cos }^2\alpha$

[proved].