જો {tan ^2}θ = 2{tan ^2}phi + 1, તો cos 2θ + {sin ^2}phi = . . .

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3.Trigonometrical Ratios, Functions and Identities

easy

જો ${\tan ^2}\theta = 2{\tan ^2}\phi + 1,$ તો $\cos 2\theta + {\sin ^2}\phi = . . .$

A

$-1$

B

$0$

C

$1$

D

એકપણ નહિ.

Solution

(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 $\phi = 0$

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

Standard 11

Mathematics

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