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  4. The value of operatornamecosec2( cot -1 x)- cot 2( operatornamecosec-1 x) is equal to

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Q. The value of $\operatorname{cosec}^2\left(\cot ^{-1} x\right)-\cot ^2\left(\operatorname{cosec}^{-1} x\right)$ is equal to

Inverse Trigonometric Functions

Solution:

$ \operatorname{cosec}^2\left(\cot ^{-1} x\right)-\cot ^2\left(\operatorname{cosec}^{-1} x\right) $
$=1+\cot ^2\left(\cot ^{-1} x\right)-\left(\operatorname{cosec}^2\left(\operatorname{cosec}^{-1} x\right)-1\right)$
$=1+x^2-\left(x^2-1\right)=2$