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  4. If θ ∈(-(π/2), 0), then the value of sec ( tan -1( cot θ)- cot -1( tan θ)) equals

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Q. If $\theta \in\left(-\frac{\pi}{2}, 0\right)$, then the value of $\sec \left(\tan ^{-1}(\cot \theta)-\cot ^{-1}(\tan \theta)\right)$ equals

Inverse Trigonometric Functions

Solution:

We have
$\tan ^{-1}(\cot \theta)-\cot ^{-1}(\tan \theta)=\left(\frac{\pi}{2}-\cot ^{-1}(\cot \theta)\right)-\left(\frac{\pi}{2}-\tan ^{-1}(\tan \theta)\right)=\tan ^{-1}(\tan \theta)-\cot ^{-1}(\cot \theta) $
$=\theta-(\pi+\theta)=-\pi \quad\left(\operatorname{as} \theta \in\left(\frac{-\pi}{2}, 0\right)\right) $
$\text { Hence } \sec \left(\tan ^{-1}(\cot \theta)-\cot ^{-1}(\tan \theta)\right)=\sec (-\pi)=\sec \pi=-1 $