Question

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

Answer 1

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=(1-\tan ^2y)/(1+\tan ^2y)`

Hence using equation (1) we get:

`\cos 2y=(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=(1-(1-2\sin ^2x))/(1+2\cos ^2x-1)\\\\\cos 2y=(1-1+2\sin ^2x)/(2\cos ^2x)\\\\\cos 2y=(2\sin ^2x)/(2\cos ^2x)\\\\\cos 2y=\tan ^2x`

( since we know that:

`\tan x=(\sin x)/(\cos x)`

)

Hence we have proved that:

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