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  4. Number of solutions of the equation 2 cot -1 2+ cos -1(3 / 5)= operatornamecosec-1 x is

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Q. Number of solutions of the equation $2 \cot ^{-1} 2+\cos ^{-1}(3 / 5)=\operatorname{cosec}^{-1} x$ is

Inverse Trigonometric Functions

Solution:

$ \tan ^{-1}(1 / 2)+\tan ^{-1}(4 / 3)=\tan ^{-1} \frac{2 \cdot \frac{1}{2}}{1-\frac{1}{4}}+\tan ^{-1} \frac{4}{3}=2 \tan ^{-1} \frac{4}{3}>\frac{\pi}{2} $
$\text { but } \operatorname{cosec}^{-1} x \in\left[-\frac{\pi}{2}, 0\right) \cup\left(0, \frac{\pi}{2}\right] \Rightarrow \text { no solution }$