Relation between Inverses of Trigonometric Functions and Their Reciprocal Functions

Q. The value of $sin\left(2ta{n}^{-1}\frac{1}{3}\right)+cos\left(ta{n}^{-1}2\sqrt{2}\right)$ is

1. $\frac{14}{15}$
2. $-\frac{14}{15}$
3. $\frac{13}{12}$
4. $\frac{12}{13}$

Q. $tan\left[\frac{1}{2}co{s}^{-1}\left(\frac{\sqrt{5}}{3}\right)\right]=$

1. $\frac{3-\sqrt{5}}{2}$
2. $\frac{3+\sqrt{5}}{2}$
3. $\frac{2}{3-\sqrt{5}}$
4. $\frac{2}{3+\sqrt{5}}$

Q. $tan[2ta{n}^{-1}\left(\frac{1}{5}\right)-\frac{\pi }{4}]=$

1. $\frac{17}{7}$
2. $-\frac{17}{7}$
3. $\frac{7}{17}$
4. $-\frac{7}{17}$

Q. If $si{n}^{-1}\frac{3}{5}+co{s}^{-1}\left(\frac{12}{13}\right)=si{n}^{-1}C,thenC=$

1. $\frac{16}{65}$
2. $\frac{65}{56}$
3. $\frac{24}{65}$
4. $\frac{56}{65}$

Q. if $2co{s}^{-1}\sqrt{\frac{1+x}{2}}=\frac{\pi }{2}$, then x=

1. $1$
2. $0$
3. $\frac{-1}{2}$
4. $\frac{1}{2}$

Q. The value of ${\cos }^{-1}\left(\cos \frac{5\pi }{3}\right)+{\sin }^{-1}\left(\sin \frac{5\pi }{3}\right)$ is

1. $0$
2. $\frac{\pi }{2}$
3. $\frac{2\pi }{3}$
4. $\frac{10\pi }{3}$

Q. The value of x for which $sin\left(co{t}^{-1}\right(1+x\left)\right)=cos\left(ta{n}^{-1}x\right)$ is:

1. 1
2. 0
3. $\frac{1}{2}$
4. $-\frac{1}{2}$

Q. If ${\tan }^{-1}\frac{4}{7}+{\tan }^{-1}\frac{4}{19}+{\tan }^{-1}\frac{4}{39}+{\tan }^{-1}\frac{4}{67}+\cdots \infty =\frac{\pi }{4}+{\cot }^{-1}k,$ where $k\in Z,$ then the value of ${}^{\prime }{k}^{\prime }$ is

Q.

$se{c}^2\left[co{t}^{-1}\left(\frac{1}{2}\right)\right]+cose{c}^2\left[ta{n}^{-1}\left(\frac{1}{3}\right)\right]=$

1. 10
   

2. 15
   

3. 25

4. 20
   

Q. If x < 0, then find the value of
$2\left({\tan }^{-1}\frac{1}{x}+{\tan }^{-1}x\right)$

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