Parthasarathi Mukhopadhyay and Manabendra Nath Mukherjee Solutions for Exercise 1: EXERCISE

Simplify: $\tan \left\{\frac{1}{2}\left({\tan }^{-1}x+{\tan }^{-1}\frac{1}{x}\right)\right\}$

Simplify: $\sin \left({\sin }^{-1}\frac{1}{3}+{\sin }^{-1}\frac{1}{3}\right)+\cos \left({\tan }^{-1}\frac{1}{2}+{\tan }^{-1}\frac{1}{2}\right)=\frac{4\sqrt{k}}{9}+\frac{3}{5}$. Find the value of $k$.

Simplify: $2\left({\sin }^{-1}\left(\cos \left({\sin }^{-1}x\right)\right)+{\cos }^{-1}\left(\sin \left({\cos }^{-1}x\right)\right)\right)$. Write the answer upto two places of decimals.

Simplify: $\sin \left[{\tan }^{-1}\frac{1-x^2}{2x}+{\sec }^{-1}\frac{1+x^2}{1-x^2}\right]$

Simplify: $\sin \left({\cos }^{-1}\left(\tan \left({\sec }^{-1}x\right)\right)\right)$. If the answer is $\sqrt{p-x^2}$, find the value of $p$.

Simplify: $\cot \left({\cos }^{-1}a+{\cos }^{-1}b\right)+\cot \left({\sin }^{-1}a+{\sin }^{-1}b\right)$

Prove that ${\sin }^{-1}\left(\cos \left({\sin }^{-1}x\right)\right)+{\cos }^{-1}\left(\sin \left({\cos }^{-1}x\right)\right)=\frac{\pi }{2}, \left|x\right|\le 1.$

Solve for $x$, ${\tan }^{-1}(x+1)-{\tan }^{-1}(x-1)={\tan }^{-1}\left(2\right)$.