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Q.
If $\cot \left(\sin ^{-1} \sqrt{\frac{13}{17}}\right)=\sin \left(\tan ^{-1} x\right)$, then $x$ is equal to
Inverse Trigonometric Functions
Solution:
Given that, $\cot \left(\sin ^{-1} \sqrt{\frac{13}{17}}\right)=\sin \left(\tan ^{-1} x\right)$ Note that $x$ must be positive.
Put $ \sin ^{-1} \sqrt{\frac{13}{17}}=\theta$.
$\therefore \text { L.H.S. }=\frac{2}{\sqrt{13}}$
Put $ \tan ^{-1} x=\phi$
$\therefore \text { R.H.S. }=\frac{ x }{\sqrt{1+ x ^2}}$
So, given equation is $\frac{2}{\sqrt{13}}=\frac{ x }{\sqrt{1+ x ^2}}$ (on squaring)
$\Rightarrow x=\frac{2}{3}(\text { As } x>0) $
