( ^ -1 (cosec ( ^ -1 a ) ) )= ( where, 0< a <1) - Vidyadip

MCQ

$\cos \left(\cot ^{-1}\left(\operatorname{cosec}\left(\cos ^{-1} a\right)\right)\right)=\ldots($ where, $0< a <1)$

✓
$\frac{1}{\sqrt{2- a ^2}}$
B
$\sqrt{3-a^2}$
C
$\sqrt{2- a ^2}$
D
$\frac{1}{\sqrt{2+a^2}}$

✓

Answer

Correct option: A.

$\frac{1}{\sqrt{2- a ^2}}$

(A) Let $\cos ^{-1} a =\theta$
$\therefore \quad \cos \theta= a$
$\therefore \quad \operatorname{cosec} \theta=\frac{1}{\sqrt{1-\cos ^2 \theta}}=\frac{1}{\sqrt{1-a^2}}$
Let $\cot ^{-1}\left(\frac{1}{\sqrt{1- a ^2}}\right)=\phi$
$\therefore \quad \cot \phi=\frac{1}{\sqrt{1-a^2}}$
$\therefore \quad \tan \phi=\sqrt{1- a ^2}$
$\therefore \quad \cos \phi=\frac{1}{\sqrt{1+\tan ^2 \phi}}=\frac{1}{\sqrt{2- a ^2}}$

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