Trigonometric Ratios of Compound Angles

Q. If $sin\alpha =\frac{1}{\sqrt{5}}andsin\beta =\frac{3}{5},then\beta -\alpha$ lies in the interval

1. 
   
2. 
3. 
   
4. 

Q.

Prove the following:
cot x cot 2x - cot 2x cot 3x - cot 3x cot x = 1

Q.

If $\sec \left(\frac{x-y}{x+y}\right)=a$, then $\frac{dy}{dx}$is equal to

1. $\frac{y}{x}$

2. $\frac{-y}{x}$

3. $\frac{x}{y}$

4. $\frac{-x}{y}$

Q.

If the angles of a triangle are in the ratio $4:1:1,$ then the ratio of the longest side to the perimeter is

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

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

3. $2:3$

4. $1:6$

Q. Express each of the following as the sum or difference of sines and cosines:
(i) 2 sin 3x cos x
(ii) 2 cos 3x sin 2x
(iii) 2 sin 4x sin 3x
(iv) 2 cos 7x cos 3x

Q.

If $\sin \alpha \sin \beta -\cos \alpha \cos \beta +1=0$, then

1. $\cot \alpha +\cot \beta =0$

2. $1+\cot \alpha \cot \beta =0$

3. $1+\tan \alpha \tan \beta =0$

4. None of these

Q. 6. Prove that sin(A+B)=sinA.cosB+cosA.sinB Sin(A-B)= Cos(A-B)= Cos(A+B)=

Q. Prove that:
(i) tan 8x − tan 6x − tan 2x = tan 8x tan 6x tan 2x
(ii) $\tan \frac{\pi }{12}+\tan \frac{\pi }{6}+\tan \frac{\pi }{12}\tan \frac{\pi }{6}=1$
(iii) tan 36° + tan 9° + tan 36° tan 9° = 1
(iv) tan 13x − tan 9x − tan 4x = tan 13x tan 9x tan 4x

Q.

What is the value of sin 105^o + sin 75^o?

1. 

2. –sin15⁰

3. 2cos15⁰

4. 

Q.

What is the value of sin4x=?

1. sin2xcos2x

2. -2sin2xcos2x

3. 2 sin2xcos2x

4. –sin2xcos2x

Q. In $\Delta ABC$, if $a,b$ and $A$ are given and there are two possibilities for the third side $c_1$ and $c_2,$ then the value of $(c_1-c_2{)}^2+(c_1+c_2{)}^2{\tan }^{2}A$ is equal to

1. $a^2$
2. $4a^2$
3. $b^2$
4. $4b^2$

Q. In a triangle ABC, if angle C is obtuse and angles A and B are given by roots of the equation tan2x + p tanx + q = 0, then the value of q is

1. greater than 1
2. less than 1
3. equal to 1
4. 0

Q.

sin ${75}^0$ =

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

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

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

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

Q.

In a triangle ABC, Let , $\angle C=\frac{\pi }{2}$, if r is the inradius and R is the circumradius of the triangle ABC, then 2(r+R) equals

1. b + c

2. a + b

3. c +a

4. a + b + c

Q. Find the value of $\text{cosec}{15}^0+\sec {15}^0$.

1. $2\sqrt{2}$
2. $\sqrt{6}$
3. $2\sqrt{6}$
4. $\sqrt{6}+\sqrt{2}$

Q. The value of †an^615^°-15†an^415^°+15†an^215^°-3 is

Q. Write the value of 2 (sin⁶ x + cos⁶ x) −3 (sin⁴ x + cos⁴ x) + 1.

Q. If $\tan A=\frac{1}{3}$ and $\sec B=\frac{13}{5}$ where $\pi <A<\frac{3\pi }{2}$, $\frac{3\pi }{2}<B<2\pi$, then $\cot (A+B)$ is equal to

1. $-\frac{27}{31}$
2. $\frac{27}{31}$
3. $-\frac{31}{27}$
4. $\frac{31}{27}$

Q. The value of ${\cos }^{4}x+{\sin }^{4}x-6{\cos }^{2}x{\sin }^{2}x$ is
(a) cos 2x
(b) sin 2x
(c) cos 4x
(d) none of these

Q. Prove that:
(i) $\frac{\sin A+\sin 3A+\sin 5A}{\cos A+\cos 3A+\cos 5A}=\tan 3A$

(ii) $\frac{\cos 3A+2\cos 5A+\cos 7A}{\cos A+2\cos 3A+\cos 5A}=\frac{\cos 5A}{\cos 3A}$

(iii) $\frac{\cos 4A+\cos 3A+\cos 2A}{\sin 4A+\sin 3A+\sin 2A}=\cot 3A$

(iv) $\frac{\sin 3A+\sin 5A+\sin 7A+\sin 9A}{\cos 3A+\cos 5A+\cos 7A+\cos 9A}=\tan 6A$

(v) $\frac{\sin 5A-\sin 7A+\sin 8A-\sin 4A}{\cos 4A+\cos 7A-\cos 5A-\cos 8A}=\cot 6A$

(vi) $\frac{\sin 5A\cos 2A-\sin 6A\cos A}{\sin A\sin 2A-\cos 2A\cos 3A}=\tan A$

(vii) $\frac{\sin 11A\sin A+\sin 7A\sin 3A}{\cos 11A\sin A+\cos 7A\sin 3A}=\tan 8A$

(viii) $\frac{\sin 3A\cos 4A-\sin A\cos 2A}{\sin 4A\sin A+\cos 6A\cos A}=\tan 2A$

(ix) $\frac{\sin A\sin 2A+\sin 3A\sin 6A}{\sin A\cos 2A+\sin 3A\cos 6A}=\tan 5A$

(x) $\frac{\sin A+2\sin 3A+\sin 5A}{\sin 3A+2\sin 5A+\sin 7A}=\frac{\sin 3A}{\sin 5A}$

(xi) $\frac{\sin \left(\mathrm{\theta }+\mathrm{\varphi }\right)-2\sin \mathrm{\theta }+\sin \left(\mathrm{\theta }-\mathrm{\varphi }\right)}{\cos \left(\mathrm{\theta }+\mathrm{\varphi }\right)-2\cos \mathrm{\theta }+\cos \left(\mathrm{\theta }-\mathrm{\varphi }\right)}=\tan \mathrm{\theta }$

Q.

If $sinx+cosecx=2$, then $si{n}^{n}x+cose{c}^{n}x$ is equal to

[UPSEAT 2002]

1. 2

2. 1

3. 

4. 

Q. If $\tan px-\tan qx=0$, then the values of θ form a series in
(a) AP
(b) GP
(c) HP
(d) none of these

Q. A value of x satisfying $\cos x+\sqrt{3}\sin x=2$ is
(a) $\frac{5\pi }{3}$

(b) $\frac{4\pi }{3}$

(c) $\frac{2\pi }{3}$​

(d) $\frac{\pi }{3}$​

Q.

Art the foot of a mountain, the angle of elevation of its summit is ${45}^{\circ }$. After ascending 1 km towards the mountain up an incline of ${30}^{\circ }$, the elevation changes to ${60}^{\circ }$ (as shown in the given figure). Fin dthe height of the mountain. [Given : $\sqrt{3}=\mathrm{1.73.}]$

Q.

Match the following
Given $sinA=\frac{2}{3}andsinB=\frac{1}{4}$
$\begin{array}{cc}\left(1\right)sin(A+B)& \left(p\right)\frac{2\sqrt{15}-\sqrt{5}}{2+5\sqrt{3}}\\ \left(2\right)Cos(A-B)& \left(q\right)\frac{55}{144}\\ \left(3\right)Tan(A-B)& \left(r\right)\frac{2\sqrt{15}+\sqrt{5}}{12}\\ \left(4\right)Sin(A+B)sin(A-B)& \left(s\right)\frac{5\sqrt{3}+2}{12}\end{array}$

1. 1-S, 2-R, 3-Q, 4-P

2. 1-R, 2-P, 3-S, 4-Q

3. 1-S, 2-R, 3-P, 4-Q

4. 1-R, 2-S, 3-P, 4-Q

Q. The expression $co{s}^2(A-B)+co{s}^{2}B-2cos(A-B)cosAcosB$ is independent of

1. A
2. B
3. Both A and B
4. None

Q. If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), then write the value of tan A tan B tan C.

Q. Prove that:
(i) tan 225° cot 405° + tan 765° cot 675° = 0
(ii) $\sin \frac{8\mathrm{\pi }}{3}\cos \frac{23\mathrm{\pi }}{6}+\cos \frac{13\mathrm{\pi }}{3}\sin \frac{35\mathrm{\pi }}{6}=\frac{1}{2}$
(iii) cos 24° + cos 55° + cos 125° + cos 204° + cos 300° = $\frac{1}{2}$
(iv) tan (−225°) cot (−405°) −tan (−765°) cot (675°) = 0
(v) cos 570° sin 510° + sin (−330°) cos (−390°) = 0
(vi) $\tan \frac{11\mathrm{\pi }}{3}-2\sin \frac{4\mathrm{\pi }}{6}-\frac{3}{4}{\mathrm{cosec}}^2\frac{\mathrm{\pi }}{4}+4{\cos }^2\frac{17\mathrm{\pi }}{6}=\frac{3-4\sqrt{3}}{2}$
(vii) $3\sin \frac{\mathrm{\pi }}{6}\sec \frac{\mathrm{\pi }}{3}-4\sin \frac{5\mathrm{\pi }}{6}\cot \frac{\mathrm{\pi }}{4}=1$

Q. ${\cos }^2{1}^{\circ }+{\cos }^2{2}^{\circ }+{\cos }^2{3}^{\circ }+....+{\cos }^2{90}^{\circ }=$

1. 0
2. 1
3. 45
4. 

Q. If the sides a, b and c of ∆ABC are in H.P., prove that ${\sin }^2\frac{A}{2},{\sin }^2\frac{B}{2}\text{and}{\sin }^2\frac{C}{2}$ are in H.P.