Relation between Inverses of Trigonometric Functions and Their Reciprocal Functions

Q. The value of ${\sin }^{-1}\left(\frac{12}{13}\right)-{\sin }^{-1}\left(\frac{3}{5}\right)$ is equal to :

1. $\frac{\pi }{2}-{\cos }^{-1}\left(\frac{9}{65}\right)$
2. $\pi -{\sin }^{-1}\left(\frac{63}{65}\right)$
3. $\pi -{\cos }^{-1}\left(\frac{33}{65}\right)$
4. $\frac{\pi }{2}-{\sin }^{-1}\left(\frac{56}{65}\right)$

Q. 15. Simplify:cot (b-c)cot (c-a)+cot (c-a)cot (a-b)+cot (a-b)cot (b-c)

Q.

If x $\in (0,\pi )$ and $sinx.co{s}^{3}x>cosx.si{n}^{3}x$ then complete set of values of x is

1. 

2. 

3. 

4. 

Q. $\cot x+\cot \left(\frac{\pi }{3}+x\right)+\cot \left(\frac{\pi }{3}-x\right)=3\cot 3x$

Q. { †ext { Solve the equations: } } { †ext { (1) } \operatorname { cot } θ - \operatorname { tan } θ = 2 } { †ext { (2) } \operatorname { tan } ^ { 2 } x = 3 \operatorname { cos } e c ^ { 2 } x - 1 } { †ext { (3) } \operatorname { sin } 5 x + \operatorname { sin } 2 x = 0 } { †ext { (4) } \operatorname { cos } ^ { 2 } x + \sqrt { 3 } = 2 ( \sqrt { 3 } + 1 ) } { †ext { (5) } \operatorname { tan } x + \operatorname { tan } 2 x + \sqrt { 3 } \operatorname { tan } x \operatorname { tan } 2 x = \sqrt { 3 } } { †ext { Attachment size should be } 5 M B †ext { or less. }

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

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

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

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

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

Q. Q.11. If $\cos (\theta -\alpha )=aand\sin (\theta -\beta )=b$, then show that ${\cos }^2\left(\alpha -\beta \right)+2ab\sin \left(\alpha -\beta \right)=a^2+b^2$