Question

Let ${\tan }^{2}x=1+2{\tan }^{2}y$ then prove that $\cos 2y=1+2\cos 2x$.

Answer 1

Given,

${\tan }^{2}x=1+2{\tan }^{2}y$ …..$\left(1\right)$

Now,

$\cos 2y$

$=\left(\frac{1-{\tan }^{2}y}{1+{\tan }^{2}y}\right)$

$=\frac{1-\frac{({\tan }^{2}x-1)}{2}}{1+\frac{({\tan }^{2}x-1)}{2}}$

$=\frac{3-{\tan }^{2}x}{1+{\tan }^{2}x}$

$=\frac{(1+{\tan }^{2}x)+2(1-{\tan }^{2}x)}{1+{\tan }^{2}x}$

$=(1+2\cos 2x)$.