Properties Derived from Trigonometric Identities

Q.

The equation ${\sin }^{-1}x=|x-a|$ will have atleast one solution if

1. $a\in [-1,1]$

2. $a\in [-\frac{\pi }{2},\frac{\pi }{2}]$

3. $a\in [1,1+\frac{\pi }{2}]$

4. $a\in [1-\frac{\pi }{2},1+\frac{\pi }{2}]$

Q. If ${\tan }^{-1}x=\frac{\pi }{10}\mathrm{for}\mathrm{some}x\in R,$
then the value of $co{t}^{-1}x\mathrm{is}$

1. $\frac{2\pi }{5}$
2. $\frac{\pi }{5}$
3. $\frac{4\pi }{5}$
4. $\frac{3\pi }{5}$

Q. The least integral value of $k$ for which $(k-2)x^2+8x+k+4>{\sin }^{-1}\left(\sin 12\right)+{\cos }^{-1}\left(\cos 12\right)$ for all $x\in R$, is

Q. The value of ${\sin }^{-1}\cos \frac{29\pi }{5}$ is $\frac{a\pi }{b}$, where $a\&b$ are coprime. Then the value of $b^2-a^2$ is

Q. If $ta{n}^{-1}\left(x\right)+ta{n}^{-1}\left(y\right)+ta{n}^{-1}\left(z\right)=\pi$, then $\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx}=$

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

Q. If $a{\sin }^{-1}x-b{\cos }^{-1}x=c,$ then $a{\sin }^{-1}x+b{\cos }^{-1}x$ is equal to

1. $0$
2. $\frac{\pi ab+c(b-a)}{a+b}$
3. $\frac{\pi }{2}$
4. $\frac{\pi ab+c(a-b)}{a+b}$

Q. Which of the following is/are true regarding the range of $f\left(x\right)={\tan }^{-1}\left[\frac{2}{\pi }(2{\tan }^{-1}x-{\sin }^{-1}x+{\cot }^{-1}x-{\cos }^{-1}x)\right]$

1. Range contains only one integer
2. Range contains more than $2$ integers
3. Range contains the element $0$
4. Range contains the element $1$ and $2$

Q. The value of ${\sec }^{-1}\left(\text{cosec}\frac{11\pi }{3}\right)$ is

1. $\frac{11\pi }{3}$
2. $\frac{11\pi }{6}$
3. $\frac{5\pi }{6}$
4. $\frac{7\pi }{6}$

Q. If x, y, z are natural numbers such that $co{t}^{-1}x+co{t}^{-1}y=co{t}^{-1}z$ then the number of ordered triplets (x, y, z) that satisfy the equation is ___

1. \N
2. 1
3. 2
4. Infinite solutions

Q. The value of ${\sin }^{-1}\left(\sin \frac{5\pi }{6}\right)$ is

1. $\frac{7\pi }{6}$
2. $\frac{\pi }{6}$
3. $\frac{5\pi }{6}$
4. $-\frac{\pi }{6}$

Q.

If $x>0,y>0,z>0,xy+yz+zx<1$ and ${\tan }^{-1}x+{\tan }^{-1}y+{\tan }^{-1}z=\mathrm{\pi }$, then $x+y+z$ is equal to.

1. $0$

2. $xyz$

3. $3xyz$

4. $\sqrt{xyz}$

Q. If $x_1,x_2,x_3,x_4$ are roots of the equation $x^4-x^{3}sin2\beta +x^{2}cos2\beta -xcos\beta -sin\beta =0$ then
$ta{n}^{-1}{x}_1+ta{n}^{-1}{x}_2+ta{n}^{-1}{x}_3+ta{n}^{-1}{x}_4=$

1. $\beta$
2. $\frac{\pi }{2}-\beta$
3. $\pi -\beta$
4. None

Q.

The maximum value of $(se{c}^{-1}x{)}^2+(cose{c}^{-1}x{)}^2$ is equal to

1. $\frac{{\pi }^2}{4}$

2. $\frac{5{\pi }^2}{4}$

3. ${\pi }^2$

4. None of these

Q. $ta{n}^{-1}1+ta{n}^{-1}2+ta{n}^{-1}3=$

1. \N
2. $\frac{\pi }{2}$
3. $\pi$
4. None

Q. The value of ${\sin }^{-1}\sin \frac{36\pi }{7}+{\cos }^{-1}\sin \frac{39\pi }{7}$ is

1. $\frac{4\pi }{7}$
2. $\frac{\pi }{7}$
3. $\frac{11\pi }{14}$
4. $\frac{3\pi }{14}$

Q. If $\theta =ta{n}^{-1}\frac{d}{1+a_1{a}_2}+ta{n}^{-1}\frac{d}{1+a_2{a}_3}+\cdots +ta{n}^{-1}\frac{d}{1+a_{n-1}{a}_n}$, where $a_1,a_2,a_3,\cdots a_n$ are in A.P. with common difference d, then $tan\theta =$

1. $\frac{(n-1)d}{1+a_1{a}_n}$
2. $\frac{(n-1)d}{a_1+a_n}$
3. $\frac{a_n-a_1}{a_n+a_1}$
4. $\frac{nd}{1+a_1{a}_n}$

Q. Find the value of x, if $5{\tan }^{-1}x+2co{t}^{-1}x=2\pi .$

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

Q. If ${\cos }^{4}x-(\lambda +2){\cos }^{2}x-(\lambda +3)=0$ has a solution, then

1. $-3\le \lambda \le 0$
2. $\lambda \ge -3$
3. $-2\le \lambda \le 0$
4. $-3\le \lambda \le -2$

Q. If $x=sin\left(2ta{n}^{-1}2\right),y=sin\left(\frac{1}{2}ta{n}^{-1}\frac{4}{3}\right)$ then

1. x = 1 – y
2. $x^2=1–y$
3. $x^2=1+y$
4. $y^2=1-x$

Q. If $co{s}^{-1}\left(\sqrt{16}x\right)+co{s}^{-1}\left(3\sqrt{3}{x}^2\right)=\frac{\pi }{2}$, then x has:

1. Only one value
2. Two values
3. three values
4. four values

Q. Find the value of $tan\frac{1}{2}[si{n}^{-1}\frac{2x}{1+x^2}+co{s}^{-1}\frac{1-y^2}{1+y^2}]$
$|x|<1,y>0andxy<1$

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

Q. The value of ${\sin }^{-1}\left(\sin \frac{5\pi }{6}\right)$ is

1. $\frac{5\pi }{6}$
2. $-\frac{\pi }{6}$
3. $\frac{\pi }{6}$
4. $\frac{7\pi }{6}$

Q. The x satisfying $si{n}^{-1}x+si{n}^{-1}(1-x)=co{s}^{-1}x$ are:

1. 1, 0
2. 1, -1
3. $0,\frac{1}{2}$
4. None

Q. $Ifsi{n}^{-1}a+si{n}^{-1}b+si{n}^{-1}c=\pi ,thenthevalueofa\sqrt{(1-a^2)}+b\sqrt{(1-b^2)}+c\sqrt{(1-c^2)}willbe$

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

Q. Find the sum of the following series:
${\tan }^{-1}\frac{1}{3}+{\tan }^{-1}\frac{2}{9}+{\tan }^{-1}\frac{4}{33}+....+{\tan }^{-1}\frac{2^{n-1}}{1+2^{2n-1}}$

1. ${\tan }^{-1}{2}^n-{\tan }^{-1}1$
2. $0$
3. $\frac{\pi }{4}$
4. ${\tan }^{-1}{2}^n-\frac{\pi }{4}$

Q. Let $f\left(x\right)=({\sin }^{-1}x{)}^2-({\cos }^{-1}x{)}^2.$ If range of $f$ equals $[\frac{a{\pi }^2}{4},\frac{b{\pi }^2}{4}]$ where $a,b\in Z,$ then the value of $b-a$ is

Q. If ${\tan }^{-1}x+{\cos }^{-1}\frac{y}{\sqrt{1+y^2}}={\sin }^{-1}\frac{3}{\sqrt{10}}$ where $x,y\in N,$ then $x+y$ can be

1. $5$
2. $9$
3. $3$
4. $10$

Q. If $\alpha$ is the only real root of the equation $x^3+b{x}^2+cx+1=0(b<c)$ then the value of $ta{n}^{-1}\alpha +ta{n}^{-1}\left(\frac{1}{\alpha }\right)$ is

1. $\frac{\pi }{2}$
2. $-\frac{\pi }{2}$
3. \N
4. can not be determined

Q. The value of $ta{n}^2\left(se{c}^{-1}3\right)+co{t}^2\left(cose{c}^{-1}4\right)$ is

1. 20
2. 21
3. 23
4. 25

Q. $sin\left(\frac{1}{2}co{s}^{-1}\frac{4}{5}\right)=$

1. $\frac{1}{\sqrt{10}}$
2. $-\frac{1}{\sqrt{10}}$
3. $\frac{1}{10}$
4. $-\frac{1}{10}$