K. C. Sinha Solutions for Chapter: Inverse Trigonometric Functions, Exercise 1: EXERCISE 7.1

Prove that : ${\sin }^{-1}x+{\sin }^{-1}y={\cos }^{-1}\left(\sqrt{1-x^2}\sqrt{1-y^2}-xy\right)$, where $x\in \left[0,1\right], y\in \left[0,1\right]$.

Prove that : $\cos \left(2{\sin }^{-1}x\right)=1-2x^2$

Prove that : ${\sin }^{-1}x+{\cos }^{-1}y={\tan }^{-1}\frac{xy+\sqrt{\left\{\left(1-x^2\right)\left(1-y^2\right)\right\}}}{y\sqrt{\left(1-x^2\right)}-x\sqrt{\left(1-y^2\right)}}$

Prove that : ${\tan }^{-1}x+{\tan }^{-1}y=\frac{1}{2}{\sin }^{-1}\left[\frac{2\left(x+y\right)\left(1-xy\right)}{\left(1+x^2\right)\left(1+y^2\right)}\right]$

If ${\sin }^{-1}x+{\sin }^{-1}y=\frac{\pi }{2}$. Prove that $x\sqrt{1-y^2}+y\sqrt{1-x^2}=1$

Solve: ${\tan }^{-1}\left(x-1\right)+{\tan }^{-1}x+{\tan }^{-1}\left(x+1\right)={\tan }^{-1}3$

Solve: ${\tan }^{-1}\left(x-1\right)+{\tan }^{-1}x+{\tan }^{-1}\left(x+1\right)={\tan }^{-1}3$

Solve: ${\tan }^{-1}\frac{x+1}{x-1}+{\tan }^{-1}\frac{x-1}{x}=\pi +{\tan }^{-1}\left(-7\right)$