Pythagorean Identities

Q. If the equation ${\sin }^{-1}\left(\sin k\right)=k-2\pi$, is possesing a real solution, then

1. sum of all possible integral values of $k=18$
2. Number of real roots for $k=3$
3. Number of real roots for $k=4$
4. $k_{max}-k_{min}=2\forall k\in Z$

Q. The number of solution(s) of the equation $3\tan (x-\frac{\pi }{12})=\tan (x+\frac{\pi }{12})$ in $A=\{x\in R:x^2-6x\le 0\}$ is

1. $2$
2. $3$
3. $1$
4. $4$

Q. The number of solution(s) of the equation $3\tan (x-\frac{\pi }{12})=\tan (x+\frac{\pi }{12})$ in $A=\{x\in R:x^2-6x\le 0\}$ is

1. $2$
2. $3$
3. $1$
4. $4$

Q. The number of solutions of the equation
${\text{cosec}}^{-1}(\sin \theta +4{\sin }^2\theta )+{\sec }^{-1}(-1+6\sin \theta )=\frac{\pi }{2}$, where $\theta \in [0,5\pi ]$ is

1. $4$
2. $9$
3. $3$
4. $6$

Q. The number of solutions of the equation
${\text{cosec}}^{-1}(\sin \theta +4{\sin }^2\theta )+{\sec }^{-1}(-1+6\sin \theta )=\frac{\pi }{2}$, where $\theta \in [0,5\pi ]$ is

1. $4$
2. $9$
3. $3$
4. $6$

Q. The number of real solution(s) of the equation $({\sin }^{-1}x{)}^3+({\cos }^{-1}x{)}^3=7({\tan }^{-1}x+{\cot }^{-1}x{)}^3$ is

Q. The number of real solution(s) of the equation $({\sin }^{-1}x{)}^3+({\cos }^{-1}x{)}^3=7({\tan }^{-1}x+{\cot }^{-1}x{)}^3$ is

Q. The value of ${\sec }^{-1}\left(\frac{1}{4}\sum _{k=0}^{10}\sec (\frac{7\pi }{12}+\frac{k\pi }{2})\sec (\frac{7\pi }{12}+\frac{(k+1)\pi }{2})\right)$ in the interval $[-\frac{\pi }{4},\frac{3\pi }{4}]$ equals

Q. The general solution of the equation $cosec\theta +\sqrt{2}=0$ is

1. $\theta =n\pi +\frac{5\pi }{4},n\in I$
2. $\theta =n\pi -\frac{5\pi }{4},n\in I$
3. $\theta =n\pi +(-1{)}^n\frac{5\pi }{4},n\in I$
4. none of these.

Q. The number of solutions of the equation
${\text{cosec}}^{-1}(\sin \theta +4{\sin }^2\theta )+{\sec }^{-1}(-1+6\sin \theta )=\frac{\pi }{2}$, where $\theta \in [0,5\pi ]$ is

1. $4$
2. $9$
3. $3$
4. $6$