Properties Derived from Trigonometric Identities

Q. Sum to infinite terms of the series $co{t}^{-1}(1^2+\frac{3}{4})+co{t}^{-1}(2^2+\frac{3}{4})+co{t}^{-1}(3^2+\frac{3}{4})+.......$ is

1. $ta{n}^{-1}2$
2. $\frac{\pi }{4}$
3. $ta{n}^{-1}3$
4. $co{t}^{-1}3$

Q.

If $si{n}^{-1}x=\theta +\beta andsi{n}^{-1}y=\theta -\beta ,then1+xy=$

1. 
2. 

3. 
   

4. 

Q.

If $co{s}^{-1}\frac{3}{5}-si{n}^{-1}\frac{4}{5}=co{s}^{-1}x$ then x equal to

1. None of these
2. 0
3. 1
4. -1

Q.

The maximum value of $(se{c}^{-1}x{)}^2+(cose{c}^{-1}x{)}^2$ is equal to

1. $\frac{{\pi }^2}{4}$

2. $\frac{5{\pi }^2}{4}$

3. ${\pi }^2$

4. None of these

Q.

If $co{s}^{-1}x>si{n}^{-1}x$ then

1. 
   

2. 
   

3. 
   

4. 
   

Q.

$ta{n}^{-1}\left[\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}}\right]=$

1. 
2. 
3. 
4. 

Q. What is the fundamental period of the function $y={\sin }^{-1}\left(\sin x\right)$.

1. $2\pi$
2. $4\pi$
3. $3\pi$
4. $\pi$

Q.

If $tan\left(co{s}^{-1}x\right)=sin\left(co{t}^{-1}\frac{1}{2}\right),$ then x =

1. 
   

2. 
3. 

4. None of these

Q. If $Si{n}^{-1}(x-\frac{x^2}{2}+\frac{x^3}{4}.....\infty )+Co{s}^{-1}(x^2-\frac{x^4}{2}+\frac{x^6}{4}....\infty )=\frac{\pi }{2}for0<|x|<\sqrt{2}$ Then x =___

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

Q. The number of real solutions of $Ta{n}^{-1}\sqrt{x(x+1)}+Si{n}^{-1}\sqrt{x^2+x+1}=\frac{\pi }{2}$ is___

1. 0
2. 1
3. infinite
4. 2