Trigonometric Equations

Q.

If $\cos \theta -\sin \theta =\sqrt{2}\sin \theta$, then $\cos \theta +\sin \theta$ is equal to

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

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

3. $2\cos \theta$

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

Q.

How do you prove $\cos 2x=2{\cos }^{2}x-1$?

Q. The number of solutions of $\sin x=\frac{|x|}{10}$ is

1. $4$
2. $6$
3. $7$
4. $8$

Q. If sin $\theta +cos\theta =0$ and $\theta$ lies in 4th quadrant then find $sin\theta$ and $cos\theta$

Q.

Complete set of values of x satisfying cos2x > |sinx|, $x\in (\frac{-\pi }{2},\pi )$ is

1. $(\frac{-\pi }{6},\frac{5\pi }{6})$

2. $(\frac{-\pi }{6},\pi )$

3. $(\frac{-\pi }{6},\frac{\pi }{6})\cup (\frac{5\pi }{6},\pi )$

4. $(\frac{\pi }{6},\frac{\pi }{2})\cup (\frac{5\pi }{6},\pi )$

Q.

If $si{n}^{2}4x+co{s}^{2}x=2sin4x.co{s}^{4}x$ then number of values of x satisfying, if $x\in [-2\pi ,2\pi ]$ is

1. 0

2. 2

3. 3

4. 4

Q.

Find the principal solution or solution of $\sin x=\frac{1}{\sqrt{2}}$.

Q. If $y=a\cos \left(\ln x\right)+b\sin \left(\ln x\right)$, then $x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}$ is equal to

1. 0
2. $y$
3. $-y$
4. none of these

Q. The value of $(\sqrt{3}+\tan 1^{\circ })(\sqrt{3}+\tan 2^{\circ })(\sqrt{3}+\tan 3^{\circ })\cdots \cdots (\sqrt{3}+\tan {28}^{\circ })(\sqrt{3}+\tan {29}^{\circ })$ is

1. $2^{28}$
2. $2^{15}+2-\sqrt{3}$
3. $2^{29}$
4. $2^{14}+2-\sqrt{3}$

Q. If $x+\sin y=2020$ and $x+2020\cos y=2019$, where $0\le y\le \frac{\pi }{2},$ then the value of $[x+y]$ is
( $[.]$ denotes the greatest integer function )

Q. If $tan\theta =\frac{sin\alpha -cos\alpha }{sin\alpha +cos\alpha },$ then show that $sin\alpha +cos\alpha =\sqrt{2}cos\theta$.

Q.

How do you prove $\cos \left(\mathrm{\pi }-\mathrm{x}\right)=-\cos x$?

Q.

General solution of $\sin x+\cos x=mi{n}_{a\in R}\left\{1,a^2-4a+6\right\}$ is

1. $\frac{n\mathrm{\pi }}{2}+{\left(-1\right)}^n\frac{\mathrm{\pi }}{4}$

2. $2n\mathrm{\pi }+{\left(-1\right)}^{\mathrm{n}}\frac{\mathrm{\pi }}{4}$

3. $n\mathrm{\pi }+{\left(-1\right)}^{\mathrm{n}+1}\frac{\mathrm{\pi }}{4}$

4. $n\mathrm{\pi }+{\left(-1\right)}^{\mathrm{n}}\frac{\mathrm{\pi }}{4}-\frac{\mathrm{\pi }}{4}$

Q.

In a triangle $ABC$, if $A=\frac{\mathrm{\pi }}{4}\text{and}\tan B\times \tan C=k,$ then $k$ must satisfy

1. $k^2+6k+1=0$

2. $k^2–6k+1<0$

3. $k^2–6k+1\ge 0$

4. None of the above

Q.

If tan A +cot A =4, then the value of $ta{n}^{4}A+co{t}^{4}A.$

Q.

If $tan\alpha =\frac{1-cos\beta }{sin\beta }$, then

1. $tan3\alpha =tan2\beta$

2. $tan2\alpha =tan\beta$

3. $tan2\beta =tan\alpha$

4. none of these

Q. If $\alpha$ is a root of equation $(2\sin x-\cos x)(1+\cos x)={\sin }^{2}x$, $\beta$ is a root of equation $3{\cos }^{2}x-10\cos x+3=0$ and $\gamma$ is a root of equation $1-\sin 2x=\cos x-\sin x;0\le \alpha ,\beta ,\gamma \le \frac{\pi }{2},$ then $\sin (\alpha -\beta )$ is equal to

1. $\frac{1-\sqrt{6}}{6}$
2. $\frac{1-2\sqrt{3}}{6}$
3. $\frac{1-2\sqrt{6}}{6}$
4. $\frac{\sqrt{3}-2\sqrt{2}}{6}$

Q.

Least positive integral value of x satisfying
$(e^x-2)(sinx-cosx)(x-lo{g}_{e}2)(cosx-\frac{1}{\sqrt{2}})<0$ is

1. 3

2. 5

3. 2

4. 4

Q. If $\sin x+\cos x=\sqrt{y^2+\frac{1}{y^2}}$ for $x\in [0,\pi ]$ has a solution, then which of the following is/are correct?

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

Q.

Trigometric identities are valid for all real values where as trigonometric equations are valid for only some values.

1. 1
2. 2

Q. If $cosx-sin\alpha cot\beta sinx=cos\alpha$ ,
then the value of $tan\left(\frac{x}{2}\right)$ is

1. $-tan\left(\frac{\alpha }{2}\right)cot\left(\frac{\beta }{2}\right)$
2. $tan\left(\frac{\alpha }{2}\right)tan\left(\frac{\beta }{2}\right)$
3. $-cot\left(\frac{\alpha }{2}\right)tan\left(\frac{\beta }{2}\right)$
4. $cot\left(\frac{\alpha }{2}\right)tan\left(\frac{\beta }{2}\right)$

Q. The number of solutions of the equation
$\sin \left(9x\right)+\sin \left(3x\right)=0$
in the closed interval $[0,2\pi ]$ is

1. $7$
2. $13$
3. $19$
4. $25$

Q.

If sin x +cosec x =2, then write the value of $si{n}^{n}x+cose{c}^{n}x.$

Q.

If $sinA+sinB=\alpha$ and $cosA+cosB=\beta$ , then write the value of $tan\left(\frac{A+B}{2}\right)$

Q.

If $sin\theta +cos\theta =0$ and $\theta$ lies in the fourth quadrant, find $sin\theta$ and $cos\theta$.

Q. If $\tan 2\alpha \cdot \tan \beta =1$ and $\tan \alpha \cdot \tan \gamma =1$ then

1. $(1+\tan \alpha )(1+\tan \left(\frac{\beta }{2}\right))=2\sin (\alpha +\gamma )$
2. $\tan \alpha +\tan \beta =\tan \gamma$
3. $\tan (\alpha +\beta )=\tan \gamma$
4. $\tan 2\gamma \tan \beta =-1$

Q. If $x+\sin y=2020$ and $x+2020\cos y=2019$, where $0\le y\le \frac{\pi }{2},$ then the value of $[x+y]$ is
([.] denotes greatest integer function )

Q. If $y={\tan }^{-1}\left(\frac{\cos x}{1+\sin x}\right),x\in (-\frac{\pi }{2},\frac{\pi }{2})$, then $\frac{dy}{dx}$ is equal to

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

Q. If $x+\sin y=2020$ and $x+2020\cos y=2019$, where $0\le y\le \frac{\pi }{2},$ then the value of $[x+y]$ is
([.] denotes greatest integer function )

Q. A solution $(x,y)$ of the system of equations $x-y=\frac{1}{3}$ and ${\cos }^2\pi x-{\sin }^2\pi y=\frac{1}{2}$ is given by

1. $(\frac{11}{6},\frac{13}{6})$
2. $(\frac{5}{4},\frac{4}{3})$
3. $(\frac{13}{6},\frac{11}{6})$
4. $(\frac{5}{12},\frac{1}{12})$