Conditional Identities

Q. Find the principal value of ${\cos }^{-1}\left(\frac{\sqrt{3}}{2}\right).$

Q.

How do you simplify $se{c}^{4}x-{\tan }^{4}x$?

Q. Find the principal value of ${\text{cosec}}^{-1}(-\sqrt{2}).$

Q.

Evaluate $\tan 5\mathrm{x}\tan 3\mathrm{x}\tan 2\mathrm{x}$

1. $\tan 5\mathrm{x}-\tan 3\mathrm{x}-\tan 2\mathrm{x}$

2. $\frac{\left(\sin 5\mathrm{x}-\sin 3\mathrm{x}-\sin 2\mathrm{x}\right)}{\left(\cos 5\mathrm{x}-\cos 3\mathrm{x}-\cos 2\mathrm{x}\right)}$

3. $0$

4. None of these

Q.

The value of $1+\cos 56°+\cos 58°-\cos 66°$is

1. $4\cos 28°\cos 29°\sin 33°$

2. $\cos 28°\cos 29°\sin 33°$

3. $4\cos 28°\sin 29°\cos 33°$

4. $4\cos 28°\sin 29°\sin 33°$

Q.

If $A+B+C=180°$, then $\frac{(\tan A+\tan B+\tan C)}{\left(\tan A\tan B\tan C\right)}$ =

1. $0$

2. $2$

3. $1$

4. $-1$

Q.

The value of $\theta$ satisfying $\cos \theta +\sqrt{3}\sin \theta =2$ is

1. ${30}^{\circ }$

2. ${60}^{\circ }$

3. ${45}^{\circ }$

4. ${90}^{\circ }$

Q.

If $A+B+C=180°$, then the value of $cot\frac{A}{2}+cot\frac{B}{2}+cot\frac{C}{2}$ will be

1. $2cot\frac{A}{2}cot\frac{B}{2}cot\frac{C}{2}$

2. $4cot\frac{A}{2}cot\frac{B}{2}cot\frac{C}{2}$

3. $cot\frac{A}{2}cot\frac{B}{2}cot\frac{C}{2}$

4. $8cot\frac{A}{2}cot\frac{B}{2}cot\frac{C}{2}$

Q.

$Ina\Delta ABC,ifa=18,b=24,c=30findcosA,cosBandcosC.$

Q.

Integrate $\int cotx\cos xdx$.

Q.

The value of $\int \frac{(\cos 2x-1)}{(\cos 2x+1)}dx$ is

1. $\tan x–x+C$

2. $\tan x+x+C$

3. $x–\tan x+C$

4. $-x–cotx+C$

Q.

Prove that, ${\sin }^{2}A+{\cos }^{2}A=1$

Q.

What is ${\tan }^{-1}x$?

Q. If in a $△ABC,$ tanA+tanB+tanC =6, then cotA cotB cotC=

1. 6
   
2. 1
   
3. none of these
4. $\frac{1}{6}$
   

Q. If in a $△ABC,$ $(\sin A+\sin B+\sin C)(\sin A+\sin B-\sin C)=3\sin A\sin B,$ then angle $C$ (in degree) is

Q.

1. λ ≠-17

2. λ ≠-18

3. λ ≠-20

4. λ ≠-19

Q.

If $A+B+C=\pi$ then, $\tan \frac{A}{2}\cdot \tan \frac{B}{2}+\tan \frac{B}{2}\cdot \tan \frac{C}{2}+\tan \frac{C}{2}\cdot \tan \frac{A}{2}$ =

1. $\tan \frac{A}{2}$ . $\tan \frac{B}{2}$ . $\tan \frac{C}{2}$

2. $\tan \frac{A}{2}$ + $\tan \frac{B}{2}$ + $\tan \frac{C}{2}$

3. $-\tan \frac{A}{2}$. $\tan \frac{B}{2}$ . $\tan \frac{C}{2}$

4. $1$

Q.

If $\tan \left(x\right)=\frac{b}{a}$ , then the value of $a\cos \left(2x\right)+b\sin \left(2x\right)$ is

1. $a$

2. $a–b$

3. $a+b$

4. $b$

Q.

If $\cos \left(9\alpha \right)=\sin \left(\alpha \right)$ and $9\alpha <90°$, then the value of $\tan \left(5\alpha \right)$ is

1. $\frac{1}{\sqrt{3}}$

2. $\sqrt{3}$

3. $1$

4. $0$

Q.

In a$∆ABC,\text{if}\angle A=\pi /2,then{\cos }^{2}B+{\cos }^{2}C$ equals to

1. $-2$

2. $-1$

3. $1$

4. $0$

Q. In a $△ABC$, if $\cos A+\cos B+\cos C=\frac{3}{2}$, then the triangle is

1. an isosceles
2. an equilateral
3. a scalene
4. a right angled

Q.

$2{\tan }^{-1}(\cos x)={\tan }^{-1}\left(\cos e{c}^{2}x\right),thenx=$

1. $\frac{\mathrm{\pi }}{2}$

2. $\mathrm{\pi }$

3. $\frac{\mathrm{\pi }}{6}$

4. $\frac{\mathrm{\pi }}{3}$

Q.

If $\alpha ,\beta$ and $\gamma \in \left[0,\frac{\pi }{2}\right]$, then the value of $\frac{\sin (\alpha +\beta +\gamma )}{\sin \alpha +\sin \beta +\sin \gamma }$ is

1. $<1$

2. $=-1$

3. $<0$

4. None of these

Q.

Find the value of $\mathrm{cosec}0°$

Q.

In a triangle $tanA+tanB+tanC=6$ and $tanAtanB=2$, then the values of $tanA$, $tanB$ and $tanC$ are

1. 1, 2, 3

2. 2, 1, 3

3. 1, 2, 0

4. none of these

Q. In a $△ABC$, if $\cos A\cos B\cos C=\frac{\sqrt{3}-1}{8}$ and $\sin A\sin B\sin C=\frac{3+\sqrt{3}}{8}$, then the angles of the triangle are

1. ${45}^{\circ },{30}^{\circ },{105}^{\circ }$
2. ${45}^{\circ },{60}^{\circ },{75}^{\circ }$
3. ${30}^{\circ },{60}^{\circ },{90}^{\circ }$
4. ${45}^{\circ },{45}^{\circ },{90}^{\circ }$

Q.

State whether the following are true or false. Justify your answer.

$\cot A$ is not defined for $A=0°$.

1. True
2. False

Q.

$\frac{d}{dx}\cos h^{-1}(secx)=?$

1. $\sec \mathrm{x}$

2. $\tan x$

3. $\sin x$

4. $\mathrm{cosec}x$

Q. In $△ABC,$ if $A,B$ and $C$ represent the angles of a triangle, then the maximum value of $\cos A+\cos B+\cos C$ is

1. $\frac{5}{2}$
2. $\frac{1}{2}$
3. $\frac{3}{2}$
4. $\frac{7}{2}$

Q. In a triangle ABC, which is not right angled, then $\sum$ cos A.cos B.cosec C =___

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