Sum of Trigonometric Ratios in Terms of Their Product

Q.

Prove that $\cos 20°·\cos 40°·\cos 60°.\cos 80°=\frac{1}{16}$

Q.

Prove that $\cos 20°\times \cos 40°\times \cos 80°=\frac{1}{8}$

Q.

How to geometrically prove that $\cos (x+y)=\cos x\cos y-\sin x\sin y$

Q.

If $\sin 2\theta +\sin 2\varphi =\frac{1}{2}$and $\cos 2\theta +\cos 2\varphi =\frac{3}{2}$ then ${\cos }^2(\theta -\varphi )$:

1. $\frac{3}{8}$

2. $\frac{5}{8}$

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

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

Q.

$cos{35}^{\circ }+cos{85}^{\circ }+cos{155}^{\circ }=$

1. 0

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

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

4. $cos{275}^{\circ }$

Q. The sum of the absolute maximum and absolute minimum values of the function $f\left(x\right)={\tan }^{-1}(\sin x-\cos x)$ in the interval $[0,\pi ]$ is

1. ${\tan }^{-1}\left(\frac{1}{\sqrt{2}}\right)-\frac{\pi }{4}$
2. 0
3. ${\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)-\frac{\pi }{4}$
4. $\frac{-\pi }{12}$

Q.

If $3^{2\sin 2\alpha -1},14$ and $3^{4-2\sin 2\alpha }$ are the first three terms of an A.P. for some $\alpha$ , then the sixth term of this A.P. is

1. $65$

2. $81$

3. $78$

4. $66$

Q.

How do you prove ${\sin }^{2}x-{\sin }^{2}y=\sin \left(x+y\right)\sin \left(x-y\right)$?

Q. If $\cos (A+B)\sin (C-D)=\cos (A-B)\sin (C+D),$ then the value of $\tan A\tan B\tan C+\tan D$ is

1. $2$
2. $1$
3. $0$
4. $-1$

Q.

The value of $cot22{\frac{1}{2}}^{\circ }$is equal to

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

2. $1+\sqrt{2}$

3. $\sqrt{2}-1$

4. None of these

Q. If $tanA=xtanB,$ Prove that $\frac{sin(A-B)}{sin(A+B)}=\frac{x-1}{x+1}$.

Q.

If $f\left(x\right)=sin\left[{\pi }^2\right]x+sin[-\pi {]}^{2}x$, where [x] denotes teh greatest integer less than or equal to x, then,

1. $f\left(\frac{\pi }{2}\right)=1$

2. $f\left(\pi \right)=2$

3. $f\left(\frac{\pi }{4}\right)=-1$

4. None of these

Q.

If 2 tan $\alpha$ = 3 tan $\beta$, then tan $(\alpha -\beta )$=

1. $\frac{sin2\beta }{5-cos2\beta }$

2. $\frac{cos2\beta }{5-cos2\beta }$

3. $\frac{sin2\beta }{5+cos2\beta }$

4. none of these

Q.

If $cosA+cosB=\frac{1}{2}$ and $sinA+sinB=\frac{1}{4}$, prove that: $tan\left(\frac{A+B}{2}\right)=\frac{1}{2}$

Q.

If $\cos \theta =-\frac{1}{2}$ and $0<\theta <{360}^{\circ }$ , then the solutions are

1. $\theta ={60}^{\circ },{240}^{\circ }$

2. $\theta ={120}^{\circ },{240}^{\circ }$

3. $\theta ={120}^{\circ },{210}^{\circ }$

4. $\theta ={120}^{\circ },{300}^{\circ }$

Q. If $\sin \theta +\sin \varphi =\sqrt{3}(\cos \varphi -\cos \theta ),$ then the value of $\sin 3\theta +\sin 3\varphi$ is
$(\text{where}\theta \in (0,{90}^{\circ }),\varphi \in (0,{90}^{\circ }\left)\right)$

Q.

$cos{40}^{\circ }+cos{80}^{\circ }+cos{160}^{\circ }+cos{240}^{\circ }=$

Q.

How do you prove $\frac{2\tan x}{1+{\tan }^{2}x}=\sin 2x$?

Q.

Write the value of $sin\frac{\pi }{12}sin\frac{5\pi }{12}$

Q.

The value of $sin{78}^{\circ }-sin{66}^{\circ }-sin{42}^{\circ }+sin{60}^{\circ }$ is

1. -1

2. None of these

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

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

Q. Find the value of $(1+\cos \frac{\pi }{8})(1+\cos \frac{3\pi }{8})(1+\cos \frac{5\pi }{8})(1+\cos \frac{7\pi }{8})$.

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

Q.

The period of the function $f\left(x\right)=\frac{[\sin \left(8x\right)\cos \left(x\right)-\sin \left(6x\right)\cos \left(3x\right)]}{[\cos \left(2x\right)\cos \left(x\right)-\sin \left(3x\right)\sin \left(4x\right)]}$

1. $\mathrm{\pi }$

2. $2\mathrm{\pi }$

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

4. None of these

Q. If $f_n\left(\theta \right)=\frac{\cos \frac{\theta }{2}+\cos 2\theta +\cos \frac{7\theta }{2}+\cdots +\cos \frac{(3n-2)\theta }{2}}{\sin \frac{\theta }{2}+\sin 2\theta +\sin \frac{7\theta }{2}+\cdots +\sin \frac{(3n-2)\theta }{2}},$ then which among the following is (are) CORRECT?

1. $f_1\left(\frac{\pi }{2}\right)=1$
2. $f_3\left(\frac{3\pi }{16}\right)=\sqrt{2}+1$
3. $f_3\left(\frac{3\pi }{16}\right)=\sqrt{2}-1$
4. $f_5\left(\frac{\pi }{28}\right)=\sqrt{2}+1$

Q.

Write the value of $\frac{sinA+sin3A}{cosA+cos3A}$

Q. The value of $\tan 6^{\circ }\tan {42}^{\circ }\tan {66}^{\circ }\tan {78}^{\circ }$ is

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

Q. The value of $\cos {50}^{\circ }+\cos {70}^{\circ }+\cos {170}^{\circ }$ is

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

Q.

If $sin2A=\lambda sin2B,provethat:\frac{tan(A+B)}{tan(A-B)}=\frac{\lambda +1}{\lambda -1}$

Q.

If $cosA=mcosB$, then write the value of $cot\frac{A+B}{2}cot\frac{A-B}{2}$

Q.

Prove that:
(i) $\frac{sinA+sin3A}{cosA-cos3A}=cotA$
(ii) $\frac{sin9A-sin7A}{cos7A-cos9A}=cot8A$
(iii) $\frac{sinA-sinB}{cosA-cosB}=tan\frac{A-B}{2}$
(iv) $\frac{sinA+sinB}{sinA-sinB}tan\left(\frac{A+B}{2}\right)cot\frac{A+B}{2}$
(v) $\frac{cosA+cosB}{cosB-cosA}=cot\frac{A+B}{2}cot\frac{A-B}{2}$

Q.

Sides of triangles are given below. Determine which of them are right triangles? In case of a right triangle, write the length of its hypotenuse.

1. $7\mathrm{cm},24\mathrm{cm},25\mathrm{cm}$

2. $3\mathrm{cm},8\mathrm{cm},6\mathrm{cm}$

3. $50\mathrm{cm},80\mathrm{cm},100\mathrm{cm}$

4. $13\mathrm{cm},12\mathrm{cm},5\mathrm{cm}$