Trigonometric Ratios of Compound Angles

Q.

If $\sin x+\cos x=\frac{1}{5}$ then $\tan \left(2x\right)$ is

1. $\frac{25}{17}$

2. $\frac{7}{25}$

3. $\frac{25}{7}$

4. $\frac{24}{7}$

Q.

Let a vertical tower $AB$have its end $A$ on the level ground. Let $C$ be the mid-point of $AB$ and $P$ be a point on the ground such that$AP=2AB$. If $\angle BPC=\beta$, then $\tan \beta$ is equal to:

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

2. $\frac{2}{9}$

3. $\frac{4}{9}$

4. $\frac{6}{7}$

Q.

Express $\sin \left(67°\right)+\cos \left(75°\right)$ in terms of trigonometric ratios of angles between $0°$ and $45°$.

Q.

If $\tan \left(A\right)=2\tan \left(B\right)+cot\left(B\right)$, then $2\tan \left(A-B\right)$ is equal to

1. $cot\left(B\right)$$\tan \left(B\right)$

2. $2\tan \left(B\right)$

3. $cot\left(B\right)$

4. $2cot\left(B\right)$

Q. If $A$ and $B$ are two positive acute angles satisfying the equations $4-3{\cos }^{2}A=2{\cos }^{2}B$ and $\cos (A+2B)=0$, then the value of $\frac{3\sin A}{2\cos B}$ is

1. $\frac{3\cos A}{\sin B}$
2. $\frac{\cos B}{\sin A}$
3. $\frac{2\cos B}{\sin B}$
4. $\frac{\sin B}{\cos A}$

Q.

If $\alpha$ and $\beta$ are the roots of the equation $6x^2-5x+1=0$, then the value of ${\tan }^{-1}\left(\alpha \right)+{\tan }^{-1}\left(\beta \right)$ is

1. $0$

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

3. $1$

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

Q.

If $\tan \left(\frac{\alpha }{2}\right),\mathrm{and}\tan \left(\frac{\beta }{2}\right)$ are the roots of $8x^2-26x+15=0$, then $\cos \left(\alpha +\beta \right)$ is equal to

1. $\frac{627}{725}$

2. $-\frac{627}{725}$

3. $-1$

4. None of these

Q. $\text{If}\sin A+{\sin }^{2}A=1\&a{\cos }^{12}A+b{\cos }^{10}A+c{\cos }^{8}A+d{\cos }^{6}A-1=0\text{then}a+b+c+d=$

1. $10$
2. $8$
3. $7$
4. $9$

Q. If $2\tan A+\cot A=\tan B$, then the value of $\cot A+2\tan (A-B)$ is

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

Q. $\mathrm{In}△ABC\mathrm{if}\cos A+\sin A-\frac{2}{\cos B+\sin B}=0,\mathrm{th}en\mathrm{the}\mathrm{triangle}\mathrm{is}\mathrm{an}$

1. Equilateral triangle
2. Isoceles triangle
3. Scalene triangle

Q. The value of the expression $\cos \frac{\pi }{15}\cos \frac{2\pi }{15}\cos \frac{4\pi }{15}\sin \frac{\pi }{30}$ is

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

Q. $\frac{1}{sin{10}^{\circ }}-\frac{\sqrt{3}}{cos{10}^{\circ }}$=

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

Q. If $\tan {40}^{\circ }+2\tan {10}^{\circ }=\cot x$, where $x\in (0,\pi /2)$, then the possible value of $x$ is

1. ${75}^{\circ }$
2. ${85}^{\circ }$
3. ${30}^{\circ }$
4. ${40}^{\circ }$

Q. The value of ${\sin }^{2}52{\frac{1}{2}}^{\circ }-{\sin }^{2}22{\frac{1}{2}}^{\circ }$ is

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

Q. If tan ${35}^{\circ }$= k, then the value of $\frac{tan{145}^{\circ }-tan{125}^{\circ }}{1+tan{145}^{\circ }tan{125}^{\circ }}$=

1. $\frac{2k}{1-k^2}$
2. $\frac{2k}{1+k^2}$
3. $\frac{1-k^2}{2k}$
4. $\frac{1-k^2}{1+k^2}$

Q. The value $\tan {100}^{\circ }+\tan {125}^{\circ }+\tan {100}^{\circ }\tan {125}^{\circ }$ is

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

Q. If $\tan A$ and $\tan B$ are the roots of $x^2-3x-7=0,$ then the value of $\sin 2(A+B)$ is

1. $\frac{48}{55}$
2. $\frac{16}{73}$
3. $\frac{48}{73}$
4. $\frac{16}{55}$

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

1. $si{n}^2\alpha$
2. $si{n}^2\beta$
3. $co{s}^2\alpha$
4. $co{s}^2\beta$

Q. If $\tan \alpha =\frac{1}{\sqrt{x(x^2+x+1)}},\tan \beta =\frac{\sqrt{x}}{\sqrt{x^2+x+1}}$
and $\tan \gamma =\sqrt{x^{-3}+x^{-2}+x^{-1}}$, where $x\ne 0$, then $\alpha +\beta$ is

1. $\gamma$
2. $2\gamma$
3. $-\gamma$
4. $-2\gamma$

Q. The value of $\frac{\cos {105}^{\circ }+\sin {105}^{\circ }}{\cos {105}^{\circ }-\sin {105}^{\circ }}$ is

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

Q. Let $A$ and $B$ (where $A>B$), be acute angles. If $\sin (A+B)=\frac{12}{13}$ and $\cos (A-B)=\frac{3}{5}$, then the value(s) of $\sin \left(2A\right)$ is/are
​​​

1. $\frac{56}{65}$
2. $\frac{16}{65}$
3. $-\frac{56}{65}$
4. $-\frac{16}{65}$

Q. If $x_1,x_2,x_3,....,x_n$ are in A.P. whose common difference is $\alpha$, then the value of $sin\alpha (sec{x}_{1}sec{x}_2+sec{x}_{2}sec{x}_3+......+sec{x}_{n-1}sec{x}_n)=$

1. $\frac{sin(n-1)\alpha }{cos{x}_{1}cos{x}_n}$
2. $\frac{sinn\alpha }{cos{x}_{1}cos{x}_n}$
3. $sin(n-1)\alpha cos{x}_{1}cos{x}_n$
4. $sinn\alpha cos{x}_{1}cos{x}_n$

Q. The value of the expression $({\tan }^{4}x+2{\tan }^{2}x+1){\cos }^{2}x$, when $x=\frac{\pi }{12}$ is equal to

1. $4(2-\sqrt{3})$
2. $4(\sqrt{2}+1)$
3. $16{\cos }^2\frac{\pi }{12}$
4. $16{\sin }^2\frac{\pi }{12}$

Q. Let $\tan A=\frac{p}{(p–1)}$ and $\tan B=\frac{1}{(2p–1)},$ if $A,B\in (0,\pi /2)$ then $A–B$ can be

1. $\frac{\pi }{2}$
2. $\frac{\pi }{4}$
3. $\frac{\pi }{3}$
4. $\frac{\pi }{6}$

Q. If $co{s}^6\alpha +si{n}^6\alpha +Ksi{n}^{2}2\alpha =1,$ then K=

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

Q. If $acos2\theta +bsin2\theta =chas\alpha and\beta$
as its solution, then the value of $tan\alpha +tan\beta$ is

1. $\frac{c+a}{2b}$
2. $\frac{2b}{c+a}$
3. $\frac{c-a}{2b}$
4. $\frac{b}{c+a}$

Q.

Match the following
Given $\sin A=\frac{2}{3}\text{and}\sin B=\frac{1}{4}$
$A$ and $B$ are acute angles.
$\begin{array}{cc}\left(1\right)\sin (A+B)& \left(p\right)\frac{2\sqrt{15}-\sqrt{5}}{2+5\sqrt{3}}\\ \left(2\right)\cos (A-B)& \left(q\right)\frac{55}{144}\\ \left(3\right)\tan (A-B)& \left(r\right)\frac{2\sqrt{15}+\sqrt{5}}{12}\\ \left(4\right)\sin (A+B)\sin (A-B)& \left(s\right)\frac{5\sqrt{3}+2}{12}\end{array}$

1. 1-r, 2-p, 3-s, 4-q

2. 1-s, 2-r, 3-q, 4-p

3. 1-s, 2-r, 3-p, 4-q

4. 1-r, 2-s, 3-p, 4-q

Q. Given that $tanA,tanB$ are the roots of the equation $x^2-px+q=0$, then the value of $si{n}^2(A+B)$ is

1. $\frac{p^2}{p^2+(1-q{)}^2}$
2. $\frac{p^2}{p^2+q^2}$
3. $\frac{q^2}{p^2-(1-q{)}^2}$
4. $\frac{p^2}{(p+q{)}^2}$

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

1. \N
2. 1
3. 45
4. $\frac{89}{2}$

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

1. $[0,\frac{\pi }{4}]$
   
2. $[\frac{\pi }{2},\frac{3\pi }{4}]$
3. $[\frac{3\pi }{4},\pi ]$
   
4. $[\pi ,\frac{5\pi }{4}]$