If ^2 = 2 ^2 + 1, then 2 + ^2 equals - Vidyadip

Question

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

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Answer

b
(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 $\phi = 0$

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

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