cos ( sin – 1 3 5 + sin – 1 5 13 + sin – 1 33 65 ) is equal to : [2025]

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Solution

Q.
$\cos (\sin^{- 1} \frac{3}{5} + \sin^{- 1} \frac{5}{13} + \sin^{- 1} \frac{33}{65})$ is equal to : [2025]

1 1

2 0

3 $\frac{32}{65}$

4 $\frac{33}{65}$

Ans.
(2)

$\cos (\sin^{- 1} \frac{3}{5} + \sin^{- 1} \frac{5}{13} + \sin^{- 1} \frac{33}{65})$

$= \cos (\tan^{- 1} \frac{3}{4} + \tan^{- 1} \frac{5}{12} + \tan^{- 1} \frac{33}{56})$

$= \cos (\tan^{- 1} \frac{(\frac{3}{4} + \frac{5}{12})}{(1 - \frac{3}{4} \cdot \frac{5}{12})} + \tan^{- 1} \frac{33}{56})$

$= \cos (\tan^{- 1} \frac{56}{33} + \tan^{- 1} \frac{33}{56})$

$= \cos [\cot^{- 1} (\frac{33}{56}) + \tan^{- 1} (\frac{33}{56})]$

$= \cos (\frac{π}{2}) = 0$ $(∵ \tan^{- 1} x + \cot^{- 1} x = \frac{π}{2})$

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