Question

If ${\tan }^2\theta =2{\tan }^2\varphi +1$, then $\cos 2\theta +2{\sin }^2\varphi =$ ?

Answer 1

We have

$\cos 2\theta =\frac{1-{\tan }^2\theta }{1+{\tan }^2\theta }=\frac{1-(2{\tan }^2\varphi +1)}{1+2{\tan }^2\varphi +1}$

${\tan }^2\theta =2{\tan }^2\varphi +1$

$=\frac{-2{\tan }^2\varphi }{2+2{\tan }^2\varphi }$

$=\frac{-{\tan }^2\varphi }{{\sec }^2\varphi }$

$=-{\sin }^2\varphi$

$\cos 2\theta =-{\sin }^2\varphi$

$\cos 2\theta +{\sin }^2\varphi =0$