Trigonometric Ratios for Sum of Two Angles

Q. If $\tan x=n\tan y,n\in R^{+}$, then maximum value of ${\sec }^2(x-y)$ is

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

Q.

$si{n}^{-1}[x\sqrt{1-x}-\sqrt{x}\sqrt{1-x^2}]=$

1. $si{n}^{-1}x-si{n}^{-1}\sqrt{1-x}$
   

2. $si{n}^{-1}x+si{n}^{-1}\sqrt{1-x}$
   

3. $si{n}^{-1}x-si{n}^{-1}\sqrt{x}$
   

4. None

Q.

$\frac{1}{tan3A-tanA}$ - $\frac{1}{cot3A-cotA}$ =

1. tanA

2. tan2A

3. cotA

4. cot2A

Q. The sum of the series ${\cot }^{-1}(2^2+\frac{1}{2})+{\cot }^{-1}(2^3+\frac{1}{2^2})+{\cot }^{-1}(2^4+\frac{1}{2^3})+......\infty$ is equal to

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

Q.

If $si{n}^{-1}\frac{1}{3}+si{n}^{-1}\frac{2}{3}=si{n}^{-1}x,$ then x is equal to
[Roorkee 1995]

1. 0
2. $\frac{\sqrt{5}-4\sqrt{2}}{9}$

3. $\frac{\sqrt{5}+4\sqrt{2}}{9}$

4. $\frac{x}{2}$

Q. A quadratic equation whose roots are $si{n}^2{18}^{\circ },co{s}^2{36}^{\circ }$ are

1. $16x^2–12x+1=0$
2. $x^2–12x+1=0$
3. $16x^2–2x–1=0$
4. None

Q.

If $cos3\theta$ = $\alpha cos\theta +\beta co{s}^3\theta$, then $(\alpha ,\beta )$ =

1. (3, 4)

2. (4, 3)

3. (-3, 4)

4. (3, -4)

Q. $$\begin{array}{cc}\text{Column I}& \text{Column II}\\ \text{a.}\sin ({410}^{\circ }-A)\cos ({400}^{\circ }+A)+\cos ({410}^{\circ }-A)\sin ({400}^{\circ }+A)& \text{p. -1}\\ \text{b.}\frac{{\cos }^2{1}^{\circ }-{\cos }^2{2}^{\circ }}{2\sin 3^{\circ }\sin 1^{\circ }}\text{is equal to}& \text{q. 1}\\ \text{c.}\sin (-{870}^{\circ })+\text{cosec}(-{660}^{\circ })+\tan (-{855}^{\circ })+2\cot \left({840}^{\circ }\right)+\cos \left({480}^{\circ }\right)+\sec \left({900}^{\circ }\right)& \text{r.}\frac{1}{2}\end{array}$$

Which of the following is the CORRECT combination ?

1. $a\to p,b\to q,c\to r$
2. $a\to r,b\to q,c\to p$
3. $a\to q,b\to r,c\to p$
4. $a\to r,b\to p,c\to q$

Q. If $\cos (\alpha +\beta )=\frac{3}{5},\sin (\alpha -\beta )=\frac{5}{13}$ and $0<\alpha ,\beta <\frac{\pi }{4},$ then $\tan \left(2\alpha \right)$ is equal to:

1. $\frac{21}{16}$
2. $\frac{63}{16}$
3. $\frac{63}{52}$
4. $\frac{33}{52}$

Q. If ${\sin }^{3}x\cos 3x+{\cos }^{3}x\sin 3x=\frac{3}{8},$ then the value of $16\sin 4x$ is

Q.

If $cos3\theta$ = $\alpha cos\theta +\beta co{s}^3\theta$, then $(\alpha ,\beta )$ =

1. (3, 4)

2. (4, 3)

3. (-3, 4)

4. (3, -4)

Q. If $\sin \theta =\frac{3}{5},\cos \varphi =\frac{12}{13}$ where $\theta ,\varphi \in (0,\pi /2)$, then the value of $\tan (\theta +\varphi )$ is

1. $\frac{8}{33}$
2. $\frac{7}{33}$
3. $\frac{33}{56}$
4. $\frac{56}{33}$

Q.

If $cos(\theta -\alpha )$ = a, $sin(\theta -\beta )$ = b,

then $co{s}^2(\alpha -\beta )$ + 2ab $sin(\alpha -\beta )$ is equal to

1. $4a^2{b}^2$

2. $a^2-b^2$

3. $a^2+b^2$

4. -$a^2{b}^2$

Q. If $\sin \theta =\frac{3}{5},\cos \varphi =\frac{12}{13}$ (where $\theta$ and $\varphi$ both belongs to $1^{\text{st}}$ quadrant), then which of the following is/are true?

1. $\cot (\theta +\varphi )=\frac{63}{65}$
2. $\tan (\theta +\varphi )=\frac{56}{33}$
3. $\sin (\theta +\varphi )=\frac{56}{65}$
4. $\cos (\theta -\varphi )=\frac{63}{65}$

Q. If $\cos \theta -\sin \theta =\frac{1}{5}$, where $0<\theta <\frac{\pi }{2}$, then the value of $5(\sin \theta +\cos \theta )$ is

Q. $\cos 6\theta$ is equal to $4{\cos }^{3}3\theta -3\cos 3\theta$

1. True
2. False

Q.

If sin A = $\frac{4}{5}$ and cos B = - $\frac{12}{13}$, where A and B lie in first

and third quadrant respectively, then cos(A + B) =

1. $\frac{56}{65}$

2. - $\frac{56}{65}$

3. $\frac{16}{65}$

4. - $\frac{16}{65}$

Q. A quadratic equation whose roots are $si{n}^2{18}^{\circ },co{s}^2{36}^{\circ }$ are

1. $16x^2–12x+1=0$
2. $x^2–12x+1=0$
3. $16x^2–2x–1=0$
4. None

Q. The value of ${\sum }_{k=1}^{13}\frac{1}{sin(\frac{\pi }{4}+\frac{(k-1)\pi )}{6})sin(\frac{\pi }{4}+\frac{k\pi }{6})}$ is equal to

1. $3-\sqrt{3}$
2. $2(3-\sqrt{3})$
3. $2(\sqrt{3}-1)$
4. $2(2-\sqrt{3})$

Q. If $\cos (4y-3x-2)-\cos (4y+3x+2)=2+2\ln (k^4-255)$ and $\cos (4y-3x-2)+\cos (4y+3x+2)=2k+8$ have real solutions $(x,y)$, then the range of $k$ is

1. $[3,4]$
2. $[-3,4]$
3. $[-4,-3]$
4. $[-4,3]$

Q. If $\tan {70}^{\circ }=\tan {20}^{\circ }+\lambda \tan {50}^{\circ },$ then $\lambda$ is equal to

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

Q. If $x$ is an acute angle such that both $\tan \left(kx\right)$ and $\cot \left(kx\right)$ is defined for $k\in (0,20),$ then which of the following is/are correct ?

1. $\tan 15x+\tan 11x+\tan 4x=\tan 15x\tan 11x\tan 4x$
2. $\tan 15x-\tan 11x-\tan 4x=\tan 15x\tan 11x\tan 4x$
3. $\cot 2x\cot 4x-\cot 4x\cot 6x-\cot 2x\cot 6x=1$
4. $\cot 4x\cot 6x-\cot 2x\cot 4x-\cot 2x\cot 6x=1$

Q. If $r:R:r_1=2:5:12,$ then $\angle A=$

1. ${60}^{\circ }$
2. ${30}^{\circ }$
3. ${45}^{\circ }$
4. ${90}^{\circ }$

Q. If $f\left(x\right)=[\sin x+[\cos x+[\tan x+[\sec x\left]\right]\left]\right]$, $x\in (0,\frac{\pi }{4}),$ where $[\cdot ]$ denotes the greatest integer less than or equal to $x$, then

1. $f$ is one-one function
2. Range of $f$ is$\{2,3\}$
3. $f$ is many-one function
4. Range of $f$ is $\{0,1\}$

Q. The value of ${\tan }^2{27}^{\circ }+2\tan {27}^{\circ }\tan {36}^{\circ }$ is equal to

Q.

If tan A = - $\frac{1}{2}$ and tan B = - $\frac{1}{3}$, then A + B =

[IIT 1967; MNR 1987; MP PET 1989]

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

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

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

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 match the entries of $\text{Column-I}$ with their corresponding values in $\text{Column-II}.$

$\begin{array}{cccc}& \text{Column-I}& & \text{Column-II}\\ \text{(A)}& f_1\left(\frac{\pi }{2}\right)& \text{(P)}& 0\\ \text{(B)}& f_3\left(\frac{3\pi }{16}\right)& \text{(Q)}& \sqrt{2}-1\\ \text{(C)}& f_4\left(\frac{2\pi }{11}\right)& \text{(R)}& \sqrt{2}+1\\ \text{(D)}& f_5\left(\frac{\pi }{28}\right)& \text{(S)}& 1\\ & & \text{(T)}& 2-\sqrt{3}\end{array}$

1. $\left(A\right)\to \left(P\right)\left(B\right)\to \left(Q\right)\left(C\right)\to \left(R\right)\left(D\right)\to \left(S\right)$
   
2. $\left(A\right)\to \left(S\right)\left(B\right)\to \left(Q\right)\left(C\right)\to \left(P\right)\left(D\right)\to \left(R\right)$
   
3. $\left(A\right)\to \left(S\right)\left(B\right)\to \left(Q\right)\left(C\right)\to \left(P\right)\left(D\right)\to \left(T\right)$
   
4. $\left(A\right)\to \left(T\right)\left(B\right)\to \left(R\right)\left(C\right)\to \left(P\right)\left(D\right)\to \left(R\right)$

Q. The value of $\sum _{n=1}^5\tan \left(\frac{\theta }{2^{n+1}}\right)\sec \left(\frac{\theta }{2^n}\right)$ is

1. $\tan \left(\theta \right)-\tan \left(\frac{\theta }{16}\right)$
2. $\tan \left(\frac{\theta }{16}\right)-\tan \left(\theta \right)$
3. $\tan \left(\theta \right)-\tan \left(\frac{\theta }{32}\right)$
4. $\tan \left(\frac{\theta }{2}\right)-\tan \left(\frac{\theta }{64}\right)$

Q.

If $tan\theta$ = $\frac{sin\alpha -cos\alpha }{sin\alpha +cos\alpha }$, then $sin\alpha +cos\alpha$ and

$sin\alpha -cos\alpha$ must be equal to

1. $\sqrt{2}cos\theta ,\sqrt{2}sin\theta$

2. $\sqrt{2}sin\theta ,\sqrt{2}cos\theta$

3. $\sqrt{2}sin\theta ,\sqrt{2}sin\theta$

4. $\sqrt{2}cos\theta ,\sqrt{2}cos\theta$

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 match the entries of $\text{Column-I}$ with their corresponding values in $\text{Column-II}.$

$\begin{array}{cccc}& \text{Column-I}& & \text{Column-II}\\ \text{(A)}& f_1\left(\frac{\pi }{2}\right)& \text{(P)}& 0\\ \text{(B)}& f_3\left(\frac{3\pi }{16}\right)& \text{(Q)}& \sqrt{2}-1\\ \text{(C)}& f_4\left(\frac{2\pi }{11}\right)& \text{(R)}& \sqrt{2}+1\\ \text{(D)}& f_5\left(\frac{\pi }{28}\right)& \text{(S)}& 1\\ & & \text{(T)}& 2-\sqrt{3}\end{array}$

1. $\left(A\right)\to \left(P\right)\left(B\right)\to \left(Q\right)\left(C\right)\to \left(R\right)\left(D\right)\to \left(S\right)$
   
2. $\left(A\right)\to \left(S\right)\left(B\right)\to \left(Q\right)\left(C\right)\to \left(P\right)\left(D\right)\to \left(R\right)$
   
3. $\left(A\right)\to \left(S\right)\left(B\right)\to \left(Q\right)\left(C\right)\to \left(P\right)\left(D\right)\to \left(T\right)$
   
4. $\left(A\right)\to \left(T\right)\left(B\right)\to \left(R\right)\left(C\right)\to \left(P\right)\left(D\right)\to \left(R\right)$