If $1+2\cot^2y=\cot^2x$, show that $\cos^2y=\cos2x$.

Question

Is the following even possible to prove?

If $1+2\cot^2y=\cot^2x$, show that $\cos^2y=\cos2x$.

Answer 1

Hint:

$$\cos2x=\dfrac{1-\tan^2x}{1+\tan^2x}=\dfrac{\cot^2x-1}{\cot^2x+1}=?$$

Answer 2

Yes it is true, it can be proved as below:

From the first relation $$\cos x= \frac{\sqrt{1+2 \cot^2 y}}{\sqrt2 \csc y}$$and $$\sin x=\frac{1}{\sqrt{2} \csc x}.$$ Next $$ \cos 2x =\cos^2 x- \sin^2 x=\frac{1+2 \cot^2 y}{2 \csc ^2 y}-\frac{1}{2 \csc^2 y}=\frac{\cot^2 y}{\csc^2 y}= \cos^2 y$$