(b) ${\tan ^2}\theta = 2{\tan ^2}\phi + 1 $

$\Rightarrow 1 + {\tan ^2}\theta = 2\,(1 + {\tan ^2}\phi )$ 

==> ${\sec ^2}\theta = 2{\sec ^2}\phi $

$\Rightarrow {\cos ^2}\phi = 2{\cos ^2}\theta $ 

==> ${\cos ^2}\phi = 1 + \cos 2\theta $

$\Rightarrow {\sin ^2}\phi + \cos 2\theta = 0$.

Trick : Let $\theta = {45^o}$, then $\phi = 0$

$\therefore \;\cos (2 \times {45^o}) + {\sin ^2}0 = 0 + 0 = 0$.