Trigonometric Functions

Q. If $\alpha ={\cos }^{-1}\left(\frac{3}{5}\right),\beta ={\tan }^{-1}\left(\frac{1}{3}\right),$ where $0<\alpha ,\beta <\frac{\pi }{2},$ then $\alpha -\beta$ is equal to:

1. ${\sin }^{-1}\left(\frac{9}{5\sqrt{10}}\right)$
2. ${\tan }^{-1}\left(\frac{9}{5\sqrt{10}}\right)$
3. ${\tan }^{-1}\left(\frac{9}{14}\right)$
4. ${\cos }^{-1}\left(\frac{9}{5\sqrt{10}}\right)$

Q. If $\sum _{n=1}^{100}{\tan }^{-1}(n^2+n+1)=\frac{a\pi }{b}-{\tan }^{-1}c,$ where $a,b,c\in N$ and $a$ and $b$ are relatively prime, then the value of $\frac{a-c}{b}$ is

Q. ${\sin }^{-1}\left(\sin \frac{2\pi }{3}\right)+{\cos }^{-1}\left(\cos \frac{7\pi }{6}\right)+{\tan }^{-1}\left(\tan \frac{3\pi }{4}\right)$ is equal to:

1. $\frac{17\pi }{12}$
2. $\frac{11\pi }{12}$
3. $\frac{31\pi }{12}$
4. $-\frac{3\pi }{4}$

Q. If $\alpha ={\cos }^{-1}\left(\frac{3}{5}\right),\beta ={\tan }^{-1}\left(\frac{1}{3}\right),$ where $0<\alpha ,\beta <\frac{\pi }{2},$ then $\alpha -\beta$ is equal to:

1. ${\tan }^{-1}\left(\frac{9}{14}\right)$
2. ${\sin }^{-1}\left(\frac{9}{5\sqrt{10}}\right)$
3. ${\cos }^{-1}\left(\frac{9}{5\sqrt{10}}\right)$
4. ${\tan }^{-1}\left(\frac{9}{5\sqrt{10}}\right)$