Question

If cos(2x) = tan^2(y), show that cos(2y) = tan^2(x).

Answer 1

Answer:

We are given that:

$\cos 2x=\tan ^2y---------(1)$

Now we are asked to show that:

$\cos 2y=\tan ^2x$

We know that:

$\cos 2y=\dfrac{1-\tan ^2y}{1+\tan ^2y}$

Hence using equation (1) we get:

$\cos 2y=\dfrac{1-\cos 2x}{1+\cos 2x}--------(2)$

Also we know that:

$\cos 2x=1-2 \sin ^2x$

and $\cos 2x=2 \cos ^2x-1$

so using above two formula in equation (2) we get:

$\cos 2y=\dfrac{1-(1-2\sin ^2x)}{1+2\cos ^2x-1}\\\\\cos 2y=\dfrac{1-1+2\sin ^2x}{2\cos ^2x}\\\\\cos 2y=\dfrac{2\sin ^2x}{2\cos ^2x}\\\\\cos 2y=\tan ^2x$

( since we know that:

$\tan x=\dfrac{\sin x}{\cos x}$

)

Hence we have proved that:

$\cos 2y=\tan ^2x$