Question

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

Answer 1

Given,

${\tan }^2\alpha =1+2{\tan }^2\beta$.

or, ${\tan }^2\beta =\frac{{\tan }^2\alpha -1}{2}$......(1).

Now $\cos 2\beta$

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

$=\frac{1-\frac{{\tan }^2\alpha -1}{2}}{1+\frac{{\tan }^2\alpha -1}{2}}$ [ Using (1)]

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

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

$=1+2\cos 2\alpha$.