Product of Trigonometric Ratios in Terms of Their Sum

Q.

If $\cos ecA+cotA=11/2$, then $\tan A=?$

1. $21/22$

2. $15/16$

3. $44/117$

4. $117/43$

Q.

The general solution of the differential equation $\frac{dy}{dx}=cotxcoty$ is

1. $\cos x=c\cos ecy$

2. $\sin x=csecy$

3. $\sin x=c\cos y$

4. $\cos x=c\sin y$

Q.

The most general solutions of the equation $secx-1=(\sqrt{2}-1)\tan x$ are given by

1. $n\mathrm{\pi }+\frac{\mathrm{\pi }}{8}$

2. $2n\mathrm{\pi },2\mathrm{n\pi }+\frac{\mathrm{\pi }}{4}$

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

4. None of these

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 $cotA=\sqrt{ac},cotB=\sqrt{\left(\frac{c}{a}\right)},cotC=\sqrt{\left(\frac{a^3}{c}\right)},c=a^2+a+1$, then

1. $A+B=C$

2. $B+C=A$

3. $C+A=B$

4. None of these

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. 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. 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. If $co{s}^{-1}\sqrt{p}+co{s}^{-1}\sqrt{1-p}+co{s}^{-1}\sqrt{1-q}=\frac{3\pi }{4}$, then the value of q is

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

Q. If $\frac{\sin (A+B)}{\sin (A-B)}=\frac{x+y}{x-y}$, then $\frac{\tan A}{\tan B}$ is equal to

1. $\frac{x}{y}$
2. $\frac{y}{x}$
3. $-\frac{y}{x}$
4. $-\frac{x}{y}$

Q. The interval(s) of $x$ which satisfy the inequality $\frac{(1-3x{)}^7(1+5x{)}^4(x+5{)}^6}{(x^2-1)(x-6{)}^3(1-2x{)}^4}>0$ is/are

1. $(-5,-1)$
2. $(-\frac{1}{5},\frac{1}{3})$
3. $(1,6)$
4. $(-\infty ,-5)$

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

1. $sin2\alpha$
2. $cos2\beta$
3. $cos2\alpha$
4. $sin2\beta$

Q. Let $f\left(x\right)=\frac{1}{x},g\left(x\right)=\frac{1}{4x^2-1}$ and $h\left(x\right)=\frac{5x}{x+2}$ be three functions and $k\left(x\right)=h\left(g\right(f\left(x\right)\left)\right).$ If domain and range of $k\left(x\right)$ are $R-\{a_1,a_2,a_3,\dots ,a_n\}$ and $R-A$ respectively, where $R$ is the set of real numbers, then

1. $n+\sum _{i=1}^n{a}_i=5$
2. $n+\sum _{i=1}^n{a}_i=10$
3. Number of integers in set $A$ is $5$
4. Number of integers in set $A$ is $7$

Q. If $\sin \theta +\sin \varphi =a$ and $\cos \theta +\cos \varphi =b$, $(a\ne b,a\ne 0,b\ne 0)$ then -

1. $\tan \theta +\tan \varphi =\frac{8ab}{(a^2+b^2{)}^2+4b^2}$
2. $\cos \theta .\cos \varphi =\frac{(a^2+b^2{)}^2-4a^2}{4(a^2+b^2)}$
3. $\cos (\theta +\varphi )=\frac{b^2-a^2}{b^2+a^2}$
4. $\sin (\theta +\varphi )=\frac{4ab}{(a^2+b^2)+2b^2}$

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. If $\sin \theta +\sin \varphi =a$ and $\cos \theta +\cos \varphi =b$, $(a\ne b,a\ne 0,b\ne 0)$ then

1. $\tan \theta +\tan \varphi =\frac{8ab}{(a^2+b^2{)}^2+4b^2}$
2. $\cos \theta \cdot \cos \varphi =\frac{(a^2+b^2{)}^2-4a^2}{4(a^2+b^2)}$
3. $\cos (\theta +\varphi )=\frac{b^2-a^2}{b^2+a^2}$
4. $\sin (\theta +\varphi )=\frac{4ab}{(a^2+b^2)+2b^2}$

Q. If $\frac{\sin (A+B)}{\sin (A-B)}=\frac{x+y}{x-y}$, then $\frac{\tan A}{\tan B}$ is equal to

1. $\frac{x}{y}$
2. $\frac{y}{x}$
3. $-\frac{y}{x}$
4. $-\frac{x}{y}$

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

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. $\frac{1+n}{1-n}\cot \alpha$