Trigonometric Ratios of Multiples of an Angle

Q.

If $y={\tan }^{-1}\left(\sqrt{\frac{a-x}{a+x}}\right)$, then $\frac{dy}{dx}=$

1. ${\cos }^{-1}\left(\frac{x}{a}\right)$

2. $-{\cos }^{-1}\left(\frac{x}{a}\right)$

3. $\frac{1}{2}{\cos }^{-1}\left(\frac{x}{a}\right)$

4. None of these

Q.

$\cos \left[{\sin }^{-1}\left(\frac{5}{13}\right)\right]=$

1. $\frac{12}{13}$

2. $-\frac{12}{13}$

3. $\frac{5}{12}$

4. None of these

Q. The value of ${\cos }^2{10}^{\circ }-\cos {10}^{\circ }\cos {50}^{\circ }+{\cos }^2{50}^{\circ }$ is equal to

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

Q.

The difference between the two acute angles of a right - angled triangle is $\frac{2\pi }{5}$ radians. Express the angles in degrees.

Q. The value of $\sin 20°\sin 40°\sin 80°$ equals

1. $\frac{\sqrt{3}}{2}$
2. $\frac{\sqrt{3}}{4}$
3. $\frac{\sqrt{3}}{8}$
4. $\frac{1}{8}$

Q. Integration of sin 4x sin x

Q. tan (cos inverse( 3/4)+sin inverse (3/4)-sec inverse (3))

Q.

If $tanA=\frac{1}{7}andtanB=\frac{1}{3}$, show that cos 2A = sin 4B.

Q.

What is the maximum value of $\frac{1}{\cos ec\theta }$?

Q.

$8sin\frac{x}{8}cos\frac{x}{2}cos\frac{x}{4}cos\frac{x}{8}$ is equal to

1. 8 cos x

2. cos x

3. 8 sin x

4. sin x

Q.

The value of $\frac{cos3\theta }{2cos2\theta -1}$ is equal to

1. $cos\theta$

2. $sin\theta$

3. $tan\theta$

4. none of these

Q. If tan(A+B)=x and tan(A-B) =y, find the values of tan2A and tan2B.

Q.

If in a triangle ABC, a=6cm, b=8cm, c=10cm, then the value of sin 2A is

1. 6/25

2. 8/25

3. 10/25

4. 24/25

Q. Let $\theta ,\varphi \in [0,2\pi ]$ be such that $2\cos \theta (1-\sin \varphi )={\sin }^2\theta (\tan \frac{\theta }{2}+\cot \frac{\theta }{2})\cos \varphi -1,\tan (2\pi -\theta )>0$ and $-1<\sin \theta <-\frac{\sqrt{3}}{2}.$ Then $\varphi$ cannot satisfy

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

Q. Diffrentiate y= sin^2(3x+4)

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

1. $0$
2. $2\pi -10$
3. $10-4\pi$
4. $3\pi -10$

Q. 25. If I(n) =integral of (sin(nx))÷(sinx) , for n>1, then the value of I(n) -I(n-2) is

Q. If A-B = $\pi /4$, then (1+tanA)(1-tanB) is equal to

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

Q.

The value of $cos\frac{\pi }{65}cos\frac{2\pi }{65}cos\frac{4\pi }{65}cos\frac{8\pi }{65}cos\frac{16\pi }{65}cos\frac{32\pi }{65}$ is

1. $\frac{1}{8}$

2. none of these

3. $\frac{1}{16}$

4. $\frac{1}{32}$

Q.

If $tan\frac{\theta }{2}=\sqrt{\frac{1-e}{1+e}}.tan\frac{\alpha }{2},\text{then}cos\alpha$ =

1. $1-ecos(cos\theta +e)$

2. $\frac{1+ecos\theta }{cos\theta -e}$

3. $\frac{1-ecos\theta }{cos\theta -e}$

4. $\frac{cos\theta -e}{1-ecos\theta }$

Q. $\tan \frac{2\pi }{5}-\tan \frac{\pi }{15}-\sqrt{3}\tan \frac{2\pi }{5}\tan \frac{\pi }{15}$ is equal to

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

Q.

Show that: $\frac{\mathrm{sinA}}{(\mathrm{secA}+\mathrm{tanA}-1)}+\frac{\mathrm{cosA}}{(\mathrm{cosecA}+\mathrm{cotA}-1)}=1.$

Q.

If $y={\tan }^{-1}\left(\frac{\sin x+\cos x}{\cos x-\sin x}\right)$, then $\frac{dy}{dx}$ is

1. $0$

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

3. $1$

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

Q.

Prove that:

$cot22{\frac{1}{2}}^{\circ }=\sqrt{2}+1$

Q.

prove that (1+cotA+tanA)(sinA-cosA)=(secA/cosec^2A)-(cosecA/sec^2A)

Q.

If $\cos ec\theta =(p+q)/(p-q)$ , then $cot\left[\right(\pi /4)+(\theta /2\left)\right]=?$

1. $\sqrt{(p/q)}$

2. $\sqrt{(q/p)}$

3. $\sqrt{pq}$

4. $pq$

Q. $\text{If}A={340}^{\text{o}},\text{then}$ $\sqrt{1-\sin A}-\sqrt{1+\sin A}$ is equal to

1. $2\sin {10}^{\text{o}}$
2. $-2\sin {10}^{\text{o}}$
3. $2\cos {10}^{\text{o}}$
4. $-2\cos {10}^{\text{o}}$

Q. If $\sin x=\frac{1}{3}$ and $\cos x<0$, then the value of $\tan 3x$ is

1. $\frac{23}{10\sqrt{2}}$
2. $-\frac{23}{10\sqrt{2}}$
3. $-\frac{23}{5\sqrt{2}}$
4. $-\frac{23}{15\sqrt{2}}$

Q. If $\cos x=\frac{2\cos y-1}{2-\cos y}$, where $x,y\in (0,\pi )$ and $\tan \frac{x}{2}\cot \frac{y}{2}=k$, then the value of $\left[k\right]$ is
(where [.] represents greatest integer function)

Q.

2(1-2 $si{n}^{2}7\theta$) $sin3\theta$ is equal to

1. $sin17\theta -11\theta$

2. $sin11\theta -17\theta$

3. $cos17\theta -cos11\theta$

4. $cos17\theta +cos11\theta$