Telangana Board Solutions for Exercise 1: Exercise 8 (a)

Prove that ${\tan }^{-1}\frac{3}{4}+{\tan }^{-1}\frac{3}{5}-{\tan }^{-1}\frac{8}{19}=\frac{\mathrm{\pi }}{4}$.

Prove that ${\tan }^{-1}\frac{1}{7}+{\tan }^{-1}\frac{1}{8}={\cot }^{-1}\frac{201}{43}+{\cot }^{-1}18$.

Show that ${\sec }^2\left({\tan }^{-1}2\right)+{\mathrm{cosec}}^2({\cot }^{-1}2)=10$.

The value of $\tan ({\cos }^{-1}\frac{4}{5}+{\tan }^{-1}\frac{2}{3})$ is $\frac{k}{6}$. Find $k$.

If ${\sin }^{-1}x-{\cos }^{-1}x=\frac{\pi }{6}$, then the value of $x$ is $\frac{\sqrt{k}}{2}$. Find $k$.

Prove that $2{\sin }^{-1}\frac{3}{5}-{\cos }^{-1}\frac{5}{13}={\cos }^{-1}\frac{323}{325}$.

Prove that ${\sin }^{-1}\frac{4}{5}+2{\tan }^{-1}\frac{1}{3}=\frac{\pi }{2}$.

If $a,b,c$ are distinct non-zero real numbers having the same sign. Prove that ${\cot }^{-1}\left(\frac{ab+1}{a-b}\right)+{\cot }^{-1}\left(\frac{bc+1}{b-c}\right)+{\cot }^{-1}\left(\frac{ca+1}{c-a}\right)=\pi \text{ or }2\pi$.