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Question

Mathematics Question on Trigonometric Identities

If cot(sin1x)=cos(tan13)cot \,(sin^{-1} \,x) = cos (tan^{-1} \sqrt 3) , then x=x =

A

00

B

23\frac{2}{\sqrt{3}}

C

22

D

25\frac{2}{\sqrt{5}}

Answer

25\frac{2}{\sqrt{5}}

Explanation

Solution

We have, cot(sin1x)cot(sin^{-1} x)
=cos(tan13)=cos\left(tan^{-1} \sqrt{3}\right)
cot(sin1x)=cos(π3)=12\Rightarrow cot \left(sin^{-1}\, x\right)=cos \left(\frac{\pi}{3}\right)=\frac{1}{2}
sin1x=cot1(12)\Rightarrow sin^{-1} x=cot^{-1}\left(\frac{1}{2}\right)
x=sin[cot1(12)]\Rightarrow x=sin\left[cot^{-1}\left(\frac{1}{2}\right)\right]
=sin[sin1(25)]=sin \left[sin^{-1}\left(\frac{2}{\sqrt{5}}\right)\right]
x=25\therefore x=\frac{2}{\sqrt{5}}
[cot1(12)=θcotθ=12sinθ=25]\left[\because cot^{-1}\left(\frac{1}{2}\right)=\theta \Rightarrow cot\,\theta=\frac{1}{2} \therefore sin \theta=\frac{2}{\sqrt{5}}\right]