Basic Trigonometric Identities

Q.

Find the value of $\frac{1}{c\mathrm{os}C}$

Q.

What is the derivative of $\sin x\cos x$?

Q.

Find the value of $\cos {18}^{\circ }$

Q.

The value of $\sin \left(600°\right)\cos \left(330°\right)+\cos \left(120°\right)\sin \left(150°\right)$ is

1. $-1$

2. $1$

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

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

Q.

Prove that:

$2(si{n}^{6}x+co{s}^{6}x)-3(si{n}^{4}x+co{s}^{4}x)+1=0$

Q. $\text{cosec}{18}^{\circ }$ is a root of the equation

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

Q.

Find six trigonometric functions of ${315}^{°}$

Q.

If tan $x=\frac{b}{a}$, then find the value of $\sqrt{\frac{a+b}{a-b}}+\sqrt{\frac{a-b}{a+b}}$.

Q.

If $g\left(f\left(x\right)\right)=\left|\sin x\right|$ and $f\left(g\left(x\right)\right)={\left(\sin \sqrt{x}\right)}^2$, then

1. $f\left(x\right)={\sin }^{2}x$ and $g\left(x\right)=\sqrt{x}$

2. $f\left(x\right)=\sin x$ and $g\left(x\right)=\left|x\right|$

3. $f\left(x\right)=x^2$ and $g\left(x\right)=\sin \sqrt{x}$

4. $f$ and $g$ can not be determined

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.

(i) If $tanA=\frac{5}{6}$ and $tanB=\frac{1}{11},$ prove that A+B= $\frac{\pi }{4}$
(ii) If $tanA=\frac{m}{m-1}$ and $tanB=\frac{m}{2m-1}$
then prove that $A-B=\frac{\pi }{4}$

Q.

If $1+cot\theta =\cos ec\theta$, then the general value of $\theta$ is

1. $n\pi +\frac{\pi }{2}$

2. $2n\pi –\frac{\pi }{2}$

3. $2n\pi +\frac{\pi }{2}$

4. None of these

Q.

The value of $\cos \left(480°\right)\sin \left(150°\right)+\sin \left(600°\right)\cos \left(390°\right)$ is

1. $0$

2. $1$

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

4. $-1$

5. $-\frac{1}{2}$

Q.

If $tan\theta =x-\frac{1}{4x}$, then sec \theta-tan \theta is equal to

1. $-2x,\frac{1}{2x}$

2. $2x,\frac{1}{2x}$

3. $-\frac{1}{2x},2x$

4. 2x

Q. For any $\theta \in (\frac{\pi }{4},\frac{\pi }{2})$, the expression $3(\sin \theta -\cos \theta {)}^4+6(\sin \theta +\cos \theta {)}^2+4{\sin }^6\theta$ equals:

1. $13-4{\cos }^2\theta +6{\sin }^2\theta {\cos }^2\theta$
2. $13-4{\cos }^6\theta$
3. $13-4{\cos }^2\theta +6{\cos }^4\theta$
4. $13-4{\cos }^4\theta +2{\sin }^2\theta {\cos }^2\theta$

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

Q.

If $x=rsin\theta cos\varphi ,y=rsin\theta sin\varphi$ and $z=rcos\theta$, then $x^2+y^2+z^2$ is independent of

1. $0,\varphi$

2. $c,\varphi$

3. $r,\theta$

4. r

Q.

The value of $co{s}^2{48}^{\circ }-si{n}^2{12}^{\circ }$ is

1. $\frac{\sqrt{5}+1}{8}$

2. $\frac{\sqrt{5}-1}{8}$

3. $\frac{\sqrt{5}+1}{5}$

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

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 $\text{cosec}\theta -\cot \theta =\frac{1}{2},$ then the value of ${\sec }^2\theta -{\cos }^2\theta$ is equal to

1. $\frac{616}{225}$
2. $\frac{706}{625}$
3. $\frac{544}{225}$
4. $\frac{44}{625}$

Q.

If $\alpha +\beta ={90}^{\circ }$, show that the maximum value of $cos\alpha cos\beta is\frac{1}{2}$

Q.

Find six trigonometric functions of ${540}^{°}$

Q. If $f\left(x\right)={\sin }^{6}x+{\cos }^{6}x,x\in R,$ then $f\left(x\right)$ lies in the interval

1. $[\frac{1}{2},1]$
2. $[\frac{1}{2},\frac{5}{8}]$
3. $[\frac{1}{4},1]$
4. $[\frac{7}{8},1]$

Q. If $\sin x+{\sin }^{2}x+{\sin }^{3}x=1$, then ${\cos }^{6}x-4{\cos }^{4}x+8{\cos }^{2}x$ is equal to

1. $4$
2. $1$
3. $0$
4. None of the above

Q.

If $ta{n}^{2}x+secx-a=0$ has alteast one solution, then $a\in$..........

1. $(-\infty ,1]$
   

2. $[1,\infty )$

3. $(-\infty ,-1]$

4. [-1, 1]

Q.

How do you prove $\sin 3\theta =3\sin \theta -4{\sin }^3\theta$?

Q.

The value of $co{s}^4+\theta +si{n}^4\theta -6co{s}^2\theta si{n}^2\theta$ is

1. $cos2\theta$

2. $sin2\theta$

3. $cos4\theta$

4. none of these

Q.

The value of$\frac{sin5\alpha -sin3\alpha }{cos5\alpha +2cos4\alpha +cos3\alpha }$ is

1. $cot\frac{\alpha }{2}$

2. none of these

3. $cot\alpha$

4. $tan\frac{\alpha }{2}$

Q. If in a triangle $△ABC,a^2{\cos }^{2}A-b^2-c^2=0,$ then

1. $\frac{\pi }{4}<A<\frac{\pi }{2}$
2. $\frac{\pi }{2}<A<\pi$
3. $A=\frac{\pi }{2}$
4. $A<\frac{\pi }{4}$

Q.

What is cosec(-585)