Subject Experts Solutions for Chapter: Inverse Trigonometric Functions, Exercise 1: Practice Exercise

What is ${\sin }^{-1}\left(\sin \frac{3\mathrm{\pi }}{5}\right)$ equal to?

If $x$ and $y$ are positive and$xy>1,$ then what is ${\tan }^{-1}x+{\tan }^{-1}y$ equal to?

What is ${\sin }^{-1}\left(\frac{3}{5}\right)+{\sin }^{-1}\left(\frac{4}{5}\right)$ equal to?

The equation ${\tan }^{-1}(1+x)+{\tan }^{-1}(1-x)=\frac{\mathrm{\pi }}{2}$ is satisfied by 

The value of $\tan \left(2{\tan }^{-1}\frac{1}{5}-\frac{\mathrm{\pi }}{4}\right)$ is 

Consider the following statements

$\mathrm{I}.$ ${\sin }^{-1}\frac{4}{5}+{\sin }^{-1}\frac{3}{5}=\frac{\pi }{2}$

$\mathrm{II}.$ ${\tan }^{-1}\sqrt{3}+{\tan }^{-1}1=-{\tan }^{-1}(2+\sqrt{3})$

Which of the above statement($s$) is / are correct?

Consider the following statements

$\mathrm{I}.$ There exists $\mathrm{\theta }\in \left(-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right)$ for which ${\tan }^{-1}\left(\tan \theta \right)\ne \theta .$

$\mathrm{II}.$ ${\sin }^{-1}\left(\frac{1}{3}\right)-{\sin }^{-1}\left(\frac{1}{5}\right)={\sin }^{-1}\left(\frac{2\sqrt{2}(\sqrt{3}-1)}{15}\right)$

Which of the above statement ($s$) is / are correct?

Consider the following statements

$\mathrm{I}.$ ${\tan }^{-1}x+{\tan }^{-1}\left(\frac{1}{x}\right)=\mathrm{\pi }$

$\mathrm{II}.$ There exist, $x, y\in [-1,1],$ where $x\ne y$ such that ${\sin }^{-1}x+{\cos }^{-1}y=\frac{\mathrm{\pi }}{2}$

Which of the above statement(s) is/are correct?