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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 $\left(\operatorname{cosec}^2\left(\cot ^{-1} x\right)-\cot ^2\left(\operatorname{cosec}^{-1} x\right)\right)$ is equal to

Inverse Trigonometric Functions

Solution:

As, $ 1+\cot ^2 \theta=\operatorname{cosec}^2 \theta $
$\text { so, } \operatorname{cosec}^2\left(\cot ^{-1} x\right)=1+\cot ^2\left(\cot ^{-1} x\right) $
$\text { and } \cot ^2\left(\operatorname{cosec}^{-1} x\right)=\operatorname{cosec}^2\left(\operatorname{cosec}^{-1} x\right)-1=\left(x^2-1\right) $
$\text { so, } \operatorname{cosec}^2\left(\cot ^{-1} x\right)-\cot ^2\left(\operatorname{cosec}^{-1} x\right)=\left(1+x^2\right)$
$\text { so, } \operatorname{cosec}^2\left(\cot ^{-1} x\right)-\cot ^2\left(\operatorname{cosec}^{-1} x\right)=\left(1+x^2\right)-\left(x^2-1\right)=1+1=2 $