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Mathematics Question on Inverse Trigonometric Functions

Write the function in the simplest form: tan⁡−11x2−1,∣x∣>1\tan^{-1}\frac{1}{\sqrt{x^2-1}},\mid{x}\mid>1

Answer

tan⁡−11x2−1,∣x∣>1\tan^{-1}\frac{1}{\sqrt{x^2-1}},\mid{x}\mid>1

put x=cosec θ ⇒\Rightarrow θ= cosec-1 x
tan⁡−11x2−1=tan⁡−11cosec⁡2θ−1\tan^{-1} \frac{1}{\sqrt{x^2-1}}=\tan^{-1}\frac{1}{\sqrt{\cosec^2\theta-1}}

tan⁡−1(1cotθ)=tan⁡−1(tan⁡θ)\tan^{-1}(\frac{1}{cot\theta})=\tan^{-1}(\tan\theta)

⇒θ=cosec⁡−1x=π2−sec⁡−1x [cosec⁡−1x+sec⁡−1x=π2]\Rightarrow \theta=\cosec^{-1}x=\frac{\pi} {2}-\sec^{-1}x\:[\cosec^{-1}x+\sec^{-1}x=\frac{\pi}{2}]