Product of Trigonometric Ratios in Terms of Their Sum

Q.

If $co{s}^{3}x.sin2x={\sum }_{r=0}^n{a}_{r}sin\left(rx\right),\forall x\in R$, then choose the correct option(s).

1. $n=5,a_1=\frac{1}{2}$

2. $n=5,a_1=\frac{1}{4}$

3. $n=5,a_3=\frac{1}{8}$

4. $n=5,a_5=\frac{1}{4}$

Q.

Prove $x=(2n\pi \pm \frac{2\pi }{3})orx=m\pi +(-1{)}^m.\frac{7\pi }{6}$, where m, $n\in I$

Q. Statement $1:$ If $xy+yz+zx=1$ where $x,y,z\in R^{+}$, then
$\frac{x}{1+x^2}+\frac{y}{1+y^2}+\frac{z}{1+z^2}=\frac{2}{\sqrt{(1+x^2)(1+y^2)(1+z^2)}}$

Statement $2:$ In a triangle $ABC$, $\sin 2A+\sin 2B+\sin 2C=4\sin A\sin B\sin C$

1. Both the statements are TRUE and Statement $2$ is the correct explanation of Statement $1$
2. Both the statements are TRUE but Statement $2$ is NOT the correct explanation of Statement $1$
3. Statement $1$ is TRUE and Statement $2$ is FALSE
4. Statement $1$ is FALSE and Statement $2$ is TRUE

Q.

If $k=sin\frac{\pi }{18}.sin\frac{5\pi }{18}.sin\frac{7\pi }{18}$ then the numerical value of k is

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

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

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

4. $Noneofthese$

Q.

$sin{20}^{\circ }sin{40}^{\circ }sin{60}^{\circ }sin{80}^{\circ }$=

1. -3/16

2. 5/16

3. 3/16

4. -5/16

Q. If $cos3x+cos2x=sin\left(\frac{3x}{2}\right)+sin\left(\frac{x}{2}\right),0\le x\le 2\pi ,thenx=$

1. 
2. 
3. 
4. 

Q.

If $k=sin\frac{\pi }{18}.sin\frac{5\pi }{18}.sin\frac{7\pi }{18}$ then the numerical value of k is

1. 

2. 

3. 

4. 

Q. If $(1-sinA)(1-sinB)(1-sinC)=(1+sinA)(1+sinB)(1+sinC)=k$

Q.

If $\alpha$ + $\beta$ - $\gamma$ = $\pi$, then $si{n}^2\alpha$ + $si{n}^2\beta$ - $si{n}^2\gamma$ =

1. 2 sin sin cos.

2. 2 cos cos cos.

3. 2 sin sin sin.

4. 2 cos cos sin.

Q.

If $\alpha +\beta -\gamma =\pi$ then $si{n}^2\alpha +si{n}^2\beta -si{n}^2\gamma$ is equal to

1. 2 sin α sin β cos γ

2. 2 sin α sin β sin γ

3. 2 cos α sin β cos γ

4. Always 0

Q. $2si{n}^2\beta +4cos(\alpha +\beta )sin\alpha sin\beta +cos2(\alpha +\beta )=$

1. 
2. 
3. 
4. 

Q. $\frac{cosA+cos3A+cos5A+cos7A}{sinA+sin3A+sin5A+sin7A}=$

1. sin 4A
2. cos 4A
3. tan 4A
4. cot 4A

Q.

Prove that $sinx+sin3x+sin5x+sin7x=4cosxcos2xsin4x.$

Q. Prove the following:
$cot4x(sin5x+sin3x)=cotx(sin5x-sin3x)$

Q. ${\cos }^{3}x\sin 2x=\sum _{r=1}^n{a}_r\sin \left(rx\right)\forall x\in R$, then

1. $n=5,a_1=\frac{1}{2}$
2. $n=5,a_1=\frac{1}{4}$
3. $n=5,a_2=\frac{1}{8}$
4. $n=5,a_2=\frac{1}{16}$

Q.

$sin{20}^{\circ }sin{40}^{\circ }sin{60}^{\circ }sin{80}^{\circ }$=

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

2. $\frac{5}{16}$

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

4. -$\frac{5}{16}$

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.

If $co{s}^{3}x.sin2x={\sum }_{x=0}^n{a}_{r}sin\left(\pi x\right),\forall x\in R$, then

1. $n=5,a_1=\frac{1}{2}$

2. $n=5,a_1=\frac{1}{4}$

3. $n=5,a_3=\frac{1}{8}$

4. $n=5,a_5=\frac{1}{4}$

Q. $(1+cos\frac{\pi }{8})(1+cos\frac{3\pi }{8})(1+cos\frac{5\pi }{8})(1+cos\frac{7\pi }{8})=$

1. 
2. 
3. 
4. 

Q.

$sin{20}^{\circ }sin{40}^{\circ }sin{60}^{\circ }sin{80}^{\circ }$=

1. -3/16

2. 5/16

3. -5/16

4. 3/16

Q.

The value of
$\sin \frac{\pi }{14}\sin \frac{3\pi }{14}\sin \frac{5\pi }{14}\sin \frac{7\pi }{14}\sin \frac{9\pi }{14}\sin \frac{11\pi }{14}\sin \frac{13\pi }{14}$ is equal to

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

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

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

4. $\frac{1}{64}$

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. $\frac{1}{256}$