Composition of Trigonometric Functions and Inverse Trigonometric Functions

Q. The value of $tan[si{n}^{-1}\left(\frac{3}{5}\right)+co{s}^{-1}\left(\frac{3}{\sqrt{13}}\right)]$ is

1. $\frac{6}{17}$
2. $\frac{6}{\sqrt{13}}$
3. $\frac{\sqrt{13}}{5}$
4. $\frac{17}{6}$

Q. The value of $\tan [2{\sin }^{-1}\frac{3}{5}+{\cos }^{-1}\frac{3}{5}]$ is

1. $\frac{4}{3}$
2. $-\frac{3}{4}$
3. $-\frac{4}{3}$
4. $\frac{3}{4}$

Q. Let $A_n=\sum _{r=1}^{2n}\sin \left({\sin }^{-1}{x}^{3r-2}\right),B_n=\sum _{r=1}^{2n}\cos \left({\cos }^{-1}{x}^{3r-1}\right),C_n=\sum _{r=1}^{2n}\tan \left({\tan }^{-1}{x}^{3r}\right),n\in N,n\ge 3$.

Which of the following option(s) is/are correct for $|x|\le 1$ ?

1. $A_n>C_n>B_n$ for $x\in (-1,1)$
2. $A_n=B_n=C_n$ has two integral roots.
3. $A_n=B_n+C_n$ has two irrational roots.
4. $B_n=C_n+A_n$ has $6n$ complex roots.

Q.

$ta{n}^{-1}\left(\frac{1}{3}\right)+ta{n}^{-1}\left(\frac{1}{7}\right)+........+ta{n}^{-1}\left(\frac{1}{n^2+n+1}\right)=$

1. $ta{n}^{-1}\left(\frac{n}{2n+1}\right)$

2. $\frac{2\pi }{10}$

3. $ta{n}^{-1}\left(\frac{n}{n+2}\right)$

4. None

Q. Evaluate:
$\sin \left({\cos }^{-1}\frac{4}{5}\right).$

1. $\frac{4}{5}$
2. $\frac{3}{4}$
3. $\frac{3}{5}$
4. Can't be determined.

Q. Evaluate:
$\cos \left(\frac{\pi }{2}-{\sin }^{-1}\left(-\frac{1}{2}\right)\right\}.$

1. $\frac{1}{2}$
2. $-\frac{1}{2}$
3. $\frac{\sqrt{3}}{2}$
4. $-\frac{\sqrt{3}}{2}$

Q. Which of the following pairs of function/functions has same graph?

1. $y=\tan \left({\cos }^{-1}x\right);y=\frac{\sqrt{1-x^2}}{x}$
2. $y=\tan \left({\cot }^{-1}x\right);y=\frac{1}{x}$
   
3. $y=\sin \left({\tan }^{-1}x\right);y=\frac{x}{\sqrt{1+x^2}}$
4. $y=\cos \left({\tan }^{-1}x\right);y=\sin \left({\cot }^{-1}x\right)$

Q. The value of $2{\tan }^{-1}\frac{1}{3}+{\tan }^{-1}\frac{1}{2}=$

1. ${30}^{\circ }$
2. ${60}^{\circ }$
3. ${\tan }^{-1}2$
4. ${\cot }^{-1}2$