Basic Trigonometric Identities

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

1. $5$
2. $25$
3. $7$
4. $23$

Q. $\frac{{\cos }^{3}A+{\sin }^{3}A}{\cos A+\sin A}+\frac{{\cos }^{3}A-{\sin }^{3}A}{\cos A-\sin A}$

1. \N
2. -1
3. 1
4. 2

Q. If $\sin \alpha ,\sin \beta ,\sin \gamma$ are in A.P and $\cos \alpha ,\cos \beta ,\cos \gamma$ are in G.P then $\frac{{\cos }^2\alpha +{\cos }^2\gamma -4\cos \alpha \cos \gamma }{1-\sin \alpha \sin \gamma }$

1. -2
2. -1
3. \N
4. 2

Q.

The value of $cot(\frac{\pi }{4}-2co{t}^{-1}3)$ is equal to

1. -7

2. None

3. 7

4. 6

Q. If $\sec x+\tan x=p$, then which of the following is/are correct?

1. $\sin x=\frac{p^2-1}{p^2+1}$
2. $\cos x=\frac{2p}{p^2+1}$
3. $\tan x=\frac{p^2-1}{2p}$
4. $\sec x=\frac{p^2+1}{2p}$

Q. If ${\tan }^{2}x=1-{\alpha }^2$, then the possible values of $\alpha$ satisfying the equation $\sec x+{\tan }^{3}x\text{cosec}x=(2-{\alpha }^2{)}^{3/2}$ is

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

Q. If $\frac{{\cos }^{4}A}{{\cos }^{2}B}+\frac{{\sin }^{4}A}{{\sin }^{2}B}=1$, then which of the following is/are correct?

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

Q. If sum of all the interior angles of a polygon is ${3060}^{\circ }$, then the number of the sides in the given polygon is

Q. If $x=3\sec \theta -2$ and $y=3\tan \theta +2$, then which of the following equation in $x,y$ is correct?

1. $x^2-y^2+4x+4y=9$
2. $x^2+y^2-2x+4y=0$
3. $x^2-y^2-4x+4y=0$
4. $x^2+y^2+2x-4y=9$

Q. If $y\cos \theta -1=\sin \theta$, then $\sin \theta$ is

1. $\sin \theta =\frac{y^2-1}{y^2+1}$
2. $\sin \theta =\frac{1}{y^2+1}$
3. $\sin \theta =\frac{y^2}{y^2+1}$
4. $\sin \theta =\frac{y-1}{y^2+1}$

Q. In $△ABC,(a-b{)}^{2}co{s}^2\frac{C}{2}+(a+b{)}^{2}si{n}^2\frac{C}{2}=$

1. $a^2$
2. $b^2$
3. $c^2$
4. None of these

Q.

$Ifaco{s}^3\alpha +3acos\alpha si{n}^2\alpha =mandasi{n}^3\alpha +3aco{s}^2\alpha sin\alpha =n,then(m+n{)}^{\frac{2}{3}}+(m-n{)}^{\frac{2}{3}}isequalto$

1. $2a^2$

2. $2a^{\frac{1}{3}}$

3. $2a^{\frac{2}{3}}$

4. $2a^3$

Q. If $(\sin \alpha +cosec\alpha {)}^2+(\cos \alpha +\sec \alpha {)}^2=k+{\tan }^2\alpha +{\cot }^2\alpha$ then k =

1. 9
2. 7
3. 5
4. 3

Q. If $P={\sec }^{6}A-{\tan }^{6}A$ and $Q=3{\sec }^{2}A{\tan }^{2}A,$ then $P-Q$ is equal to

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

Q. If $tan\theta =\sqrt{\frac{3}{2}}$, the sum of the infinite series 1 + 2 $(1-cos\theta )+3(1-cos\theta {)}^2+4(1-cos\theta {)}^3+....\infty$ is

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

Q. A sector $OABO$ of central angle $\theta$ is constructed in a circle with centre $O$ and radius $6.$ The radius of the circle that is circumscribed about the triangle $OAB,$ is

1. $6\cos \frac{\theta }{2}$
2. $6\sec \frac{\theta }{2}$
3. $6(\cos \frac{\theta }{2}+2)$
4. $3\sec \frac{\theta }{2}$

Q. If $\tan a,\tan b$ are the roots of $x^2-px+q=0$ and $\cot a,\cot b$ are the roots of $x^2-rx+s=0$, then the value of $rs$ is

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

Q. If $x=3\sec \theta -2$ and $y=3\tan \theta +2$, then which of the following equation in $x,y$ is correct?

1. $x^2-y^2+4x+4y=9$
2. $x^2+y^2-2x+4y=0$
3. $x^2-y^2-4x+4y=0$
4. $x^2+y^2+2x-4y=9$

Q. If 0 < x < $\pi$, and cos x + sin x = $\frac{1}{2}$, then tan x =

1. $\frac{-(4+\sqrt{7})}{3}$
   
2. $\frac{(1+\sqrt{7})}{4}$
   
3. $\frac{(1-\sqrt{7})}{4}$
   
4. $\frac{(4-\sqrt{7})}{4}$

Q. Consider the real valued function $f\left(x\right)=4\sin x+{\cos }^{2}x.$ Then which of the following statements is (are) FALSE?

1. Range of $f$ is $[-4,4]$
2. $f\left(\frac{\pi }{2}\right)-f\left(0\right)=4$
3. Graph of $y=f\left(x\right)$ intersects $x$-axis at only one point in the interval $[0,\pi ]$
4. $f(-x)=f\left(x\right)\Rightarrow x=n\pi ,$ where $n\in Z$

Q.

The value of the expression $1-\frac{si{n}^{2}y}{1+cosy}+\frac{1+cosy}{siny}-\frac{siny}{1-cosy}$

1. 0

2. 1

3. sin y

4. cos y

Q.

If $(\sin x+\cos x=a)$, then $1+2\sin x\cos x=a^2$

1. True

2. False

Q.

$3[si{n}^4(\frac{3\pi }{2}-\alpha )+si{n}^4(3\pi +\alpha \left)\right]-2[si{n}^6(\frac{\pi }{2}+\alpha )+si{n}^6(5\pi -\alpha \left)\right]=$

1. $0$

2. $1$

3. $3$

4. $sin4\alpha +sin6\alpha$

Q. Let $\frac{\sin x}{a}=\frac{\cos x}{b}=\frac{\tan x}{c}=2,$ where $0<x<\frac{\pi }{2}$ and $R=bc+\frac{1}{2c}+\frac{2a}{1+2b}.$ Then the minimum value of $R$ is

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

Q.

Find the value of $\frac{{\sec }^{2}x-{\text{cosec}}^{2}x}{{\tan }^{2}x-{\cot }^{2}x}$,
$(x\in (0,\frac{\pi }{2}),x\ne \frac{\pi }{4})$

Q. If $P_n={\cos }^{n}x+{\sin }^{n}x,$ then $2P_6-3P_4+1=$

1. 2
2. 3
3. \N
4. 1

Q. In a triangle $ABC,AD$ and $BE$ are medians drawn to $BC$ and $CA$ respectively. Given $AD=4,\angle DAB=\frac{\pi }{6}$ and$\angle ABE=\frac{\pi }{3}.$ If the area of triangle $ABC$ is $\frac{p}{\sqrt{3}q},$ where $p$ and $q$ are co-prime, then the value of $p+q$ is

Q. If $\alpha ,\beta$ are supplementary angles, then ${\sin }^2\alpha +{\cos }^2\beta$ =

1. 1
2. -1
3. 2
4. \N

Q. If $\alpha$ is one of the roots of $x^2+7x+10=0$, then the value of ${\cot }^{-1}\alpha +{\cot }^{-1}\frac{1}{\alpha }$ is equal to

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

Q. If $10{\sin }^{4}x+15{\cos }^{4}x=6$, then the numerical value of $24{\sec }^{6}x-27{\text{cosec}}^{6}x$ is