Product of Trigonometric Ratios in Terms of Their Sum

Q. The value of $\sin {10}^{\circ }\sin {30}^{\circ }\sin {50}^{\circ }\sin {70}^{\circ }$ is :

1. $\frac{1}{32}$
2. $\frac{1}{16}$
3. $\frac{1}{18}$
4. $\frac{1}{36}$

Q.

If $y=co{t}^{-1}\left[\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right]$, then $\frac{dy}{dx}$ is equal to ?

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

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

3. $3$

4. $1$

Q.

If $\frac{cos(A-B)}{cos(A+B)}+\frac{cos(C+D)}{cos(C-D)}=0$, prove that $tanAtanBtanCtanD=-1$

Q. Find the value of ${\tan }^{-1}\left[2\cos \left(2{\sin }^{-1}\frac{1}{2}\right)\right].$

Q.

$sin{6}^{\circ }sin{42}^{\circ }sin{66}^{\circ }sin{78}^{\circ }=\frac{1}{16}$

Q.

$cos{6}^{\circ }cos{42}^{\circ }cos{66}^{\circ }cos{78}^{\circ }=\frac{1}{16}$

Q. The value of $\cos \frac{\pi }{2^2}\cdot \cos \frac{\pi }{2^3}\cdot ...\cdot \cos \frac{\pi }{2^{10}}\cdot \sin \frac{\pi }{2^{10}}$ is :

1. $\frac{1}{1024}$
2. $\frac{1}{2}$
3. $\frac{1}{512}$
4. $\frac{1}{256}$

Q. Find an anti derivative (or integral) of the given function by the method of inspection.
$\sin 2x$

Q.

If $u=\sqrt{a^{2}co{s}^2\theta +b^{2}si{n}^2\theta }+\sqrt{a^{2}si{n}^2\theta +b^{2}co{s}^2\theta }$, then the difference between the maximum and minimum values of $u^2$ is given by

1. 

2. 

3. 

4. 

Q.

$cos{7}^{\circ }cos{14}^{\circ }cos{28}^{\circ }cos{56}^{\circ }=\frac{sin{68}^{\circ }}{16cos{83}^{\circ }}$

Q. If $x=cos\theta +isin\theta andy=cos\varphi +isin\varphi ,then{x}^m{y}^n+x^{-m}{y}^{-n}$ , is equal to

1. 
2. 
3. 
4. 

Q. If in a $△ABC$, $\tan \frac{A}{2},\tan \frac{B}{2},\tan \frac{C}{2}$ are in H.P., then the minimum value of $\cot \frac{B}{2}$ is

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

Q. If $\sin \theta =n\sin (\theta +2\alpha )$, then the value of $\tan (\theta +\alpha )$ is

1. $\frac{1-n}{1+n}\cot \alpha$
2. $\frac{1-n}{1+n}\tan \alpha$
3. $\frac{1+n}{1-n}\tan \alpha$
4. $\frac{1+n}{1-n}\cot \alpha$

Q. The value of $\frac{\sin 1^{\circ }+\sin 3^{\circ }+\sin 5^{\circ }+\sin 7^{\circ }}{\cos 1^{\circ }\cos 2^{\circ }\sin 4^{\circ }}$ is

Q. The value of ${\sec }^2{10}^{\circ }+{cosec}^2{20}^{\circ }+{cosec}^2{40}^{\circ }$ is

Q. The value of ${\sin }^{-1}x+{\sin }^{-1}\frac{1}{x}+{\cos }^{-1}x+{\cos }^{-1}\frac{1}{x}$ where ever defined is

1. $\begin{array}{c}\pi \end{array}$
2. $\begin{array}{c}\frac{\pi }{2}\end{array}$
3. $\begin{array}{c}\frac{3\pi }{2}\end{array}$
4. $2\pi$

Q.

Write the value of the expression
$\frac{1-4sin{10}^{\circ }sin{70}^{\circ }}{2sin{10}^{\circ }}$

Q. The value of $\sin \frac{17\pi }{36}\cos \frac{11\pi }{36}\cos \frac{13\pi }{36}\sin \frac{11\pi }{36}\sin \frac{13\pi }{36}\cos \frac{17\pi }{36}$ is

1. $\frac{1}{4}$
2. $\frac{1}{2}$
3. $\frac{1}{16}$
4. $\frac{1}{64}$

Q. Prove that: $\sin (n+1)x\cdot \sin (n+2)x+\cos (n+1)x\cdot \cos (n+2)x=\cos x$

Q.

The value of tan ${130}^{\circ }tan{140}^{\circ }$ is equal to

1. 

2. 

3. 

4. 

Q. The value of $\tan {20}^{\circ }\tan {80}^{\circ }\cot {50}^{\circ }$ is

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

Q.

Show that:
(i) $sin{50}^{\circ }cos{85}^{\circ }=\frac{1-\sqrt{2}sin{35}^{\circ }}{2\sqrt{2}}$
(ii) $sin{25}^{\circ }cos{115}^{\circ }=\frac{1}{2}(sin{140}^{\circ }-1)$

Q. The value of the expression $\frac{\sin 2x\cos 3x+\sin 3x\cos 8x+\sin 5x\cos 16x}{\sin 2x\sin 3x+\sin 3x\sin 8x+\sin 5x\sin 16x}$ is

1. $\cot 11x$
2. $\tan 11x$
3. $\cot 10x$
4. $\tan 10x$

Q. Find an anti derivative (or integral) of the given function by the method of inspection.
$\sin 2x-4e^{3x}$

Q. Find the values of other five trigonometric functions if $\cos x=-\frac{1}{2},x$ lies in third quadrant

Q. If ${\tan }^{-1}\left(\frac{x-1}{x-2}\right)+{\tan }^{-1}\left(\frac{x+1}{x+2}\right)=\frac{\pi }{4}$, then find the value of $x.$

Q. If $\sin 2A+\sin 2B=\frac{1}{2}$ and $\cos 2A+\cos 2B=\frac{3}{2}$, then the value of $|\cos (A-B)|$ is

1. $\sqrt{\frac{5}{8}}$
2. $\sqrt{\frac{3}{8}}$
3. $\frac{5}{8}$
4. $\frac{3}{8}$

Q. In $\Delta ABC$, if $3\tan \frac{A}{2}\tan \frac{C}{2}=1$, then $a,b,c$ are in:

1. $A.P$
2. $G.P$
3. $H.P$
4. $A.G.P$

Q. $\frac{\sin A+\sin 2A+\sin 4A+\sin 5A}{\cos A+\cos 2A+\cos 4A+\cos 5A}=$

1. $\tan 2A$
2. $\cot 3A$
3. $\tan 3A$
4. $\tan 4A$

Q. If $A={45}^{\circ },$ then the value of ${\cos }^{2}B+{\sin }^2(A+B)+2\sin A\sin ({180}^{\circ }+B)\cos ({360}^{\circ }+A+B)$ is

1. $0$
2. $\frac{1}{2}$
3. $\frac{3}{2}$
4. $2$