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Question: \(\cos \left( \sin ^ { - 1 } \frac { 5 } { 13 } \right) =\)...

cos⁡(sin⁡−1513)=\cos \left( \sin ^ { - 1 } \frac { 5 } { 13 } \right) =

A

1213\frac { 12 } { 13 }

B

−1213- \frac { 12 } { 13 }

C

512\frac { 5 } { 12 }

D

None of these

Answer

1213\frac { 12 } { 13 }

Explanation

Solution

Let sin⁡−1513=x⇒sin⁡x=513\sin ^ { - 1 } \frac { 5 } { 13 } = x \Rightarrow \sin x = \frac { 5 } { 13 }

⇒cos⁡x=1−25169=1213\Rightarrow \cos x = \sqrt { 1 - \frac { 25 } { 169 } } = \frac { 12 } { 13 }

⇒cos⁡(sin⁡−1513)=cos⁡(cos⁡−11213)=1213\Rightarrow \cos \left( \sin ^ { - 1 } \frac { 5 } { 13 } \right) = \cos \left( \cos ^ { - 1 } \frac { 12 } { 13 } \right) = \frac { 12 } { 13 }.