If ^2 A=1+2 ^2 B, then ^2 A ^2 B = - Vidyadip

MCQ

If $\tan ^2 A=1+2 \tan ^2 B$, then $\frac{\cos ^2 A}{\cos ^2 B}=$

A
$1$
✓
$\frac{1}{2}$
C
$\frac{1}{4}$
D
$\frac{\sqrt{3}}{2}$

✓

Answer

Correct option: B.

$\frac{1}{2}$

We have, $\tan ^2 A=1+2 \tan ^2 B$
$\Rightarrow \sec ^2 A-1=1+2\left(\sec ^2 B-1\right) \ \left[\because \tan ^2 \theta=\sec ^2 \theta-1\right]$
$\Rightarrow \sec ^2 A=2+2 \sec ^2 B-2 $
$\Rightarrow \sec ^2 A=2 \sec ^2 B$
$\Rightarrow \frac{1}{\cos ^2 A}=\frac{2}{\cos ^2 B} $
$\Rightarrow \frac{\cos ^2 A}{\cos ^2 B}=\frac{1}{2}$

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