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  4. For any real number x ≥ 1 the expression sec 2( tan -1 x)- tan 2( sec -1 x) is equal to

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Q. For any real number $x \geq 1$ the expression $\sec ^2\left(\tan ^{-1} x\right)-\tan ^2\left(\sec ^{-1} x\right)$ is equal to

Inverse Trigonometric Functions

Solution:

$ 1+\tan ^2\left(\tan ^{-1} x\right)-\left(\sec ^2\left(\sec ^{-1} x\right)-1\right) $
$1+\left(\tan \left(\tan ^{-1} x\right)\right)^2-\left(\sec \left(\sec ^{-1} x\right)\right)^2+1=1+x^2-x^2+1=2$