Question

Write the function tan^−1 1/√(x^2−1),| x|>1, in the simplest form.

Answer 1

Given the function $\tan ^{-1}(\frac{1}{\sqrt{x^2-1}}) ,|x|>1$.

let $x=\sec \theta$. Then

$\frac{1}{\sqrt{x^2-1}}=\frac{1}{\sqrt{\sec^2 \theta-1}}\\ \frac{1}{\sqrt{x^2-1}}=\frac{1}{\sqrt{\tan^2 \theta-1}}\\ \frac{1}{\sqrt{x^2-1}}=\frac{1}{\tan \theta}\\ \frac{1}{\sqrt{x^2-1}}=\cot \theta$

So the given function is

$\tan ^{-1}(\frac{1}{\sqrt{x^2-1}})= \tan ^{-1}(\cot \theta)\\ \tan ^{-1}(\frac{1}{\sqrt{x^2-1}})= \tan ^{-1}(\tan(\frac{\pi}{2}- \theta)) \\ \tan ^{-1}(\frac{1}{\sqrt{x^2-1}})= \frac{\pi}{2}- \theta\\ \tan ^{-1}(\frac{1}{\sqrt{x^2-1}})= \frac{\pi}{2}- \sec^{-1} x \\ \tan ^{-1}(\frac{1}{\sqrt{x^2-1}})= \csc^{-1} x ,|x|>1$