Trigonometric Identities

Q.

$1+co{t}^2\left({\sin }^{-1}x\right)=$

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

2. $x^2$

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

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

Q.

Prove that $\frac{\tan A}{1-cotA}+\frac{cotA}{1-\tan A}=1+\tan A+cotA$ .

Q. Question 1
If $cosec\theta +cot\theta =p$, then prove that $cos\theta =\frac{p^2-1}{p^2+1}$.

Q.

Prove:$\frac{\cot A+cosecA-1}{\cot A-cosecA+1}=\frac{1+\cos A}{\sin A}$

Q.

$(cosecA-sinA)(secA-cosA)=\frac{1}{(tanA+cotA)}$

1. True

2. False

Q.

If $3sin\theta +4cos\theta =5$, then the value of $sin\theta$ is _____.

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

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

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

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

Q.

Express the ratios $\sin A$ and $\cos A$ in terms of $\tan A$.

1. $\frac{\tan A}{\sqrt{1+{\tan }^{2}A}},\frac{1}{\sqrt{1+{\tan }^{2}A}}$

2. $\frac{1}{\sqrt{1+{\tan }^{2}A}},\frac{\tan A}{\sqrt{1+{\tan }^{2}A}}$

3. $\frac{\tan A}{\sqrt{1-{\tan }^{2}A}},\frac{1}{\sqrt{1-{\tan }^{2}A}}$

4. $\frac{1}{\sqrt{1-{\tan }^{2}A}},\frac{\tan A}{\sqrt{1-{\tan }^{2}A}}$

Q.

If (tan $\theta$ + cot $\theta$) = 5 then ($ta{n}^2\theta$ + $co{t}^2\theta$) = ?

(a) 27 (b) 25 (c) 24 (d) 23

Q.

If $\sec \theta -\tan \theta =\frac{1}{3}$, the value of $\sec \theta +\tan \theta$ is:

1. 1

2. 2

3. 3

4. 4

Q.

If $\tan \left(2\theta \right)\tan \left(\theta \right)=1$, then the general value of $\theta$ is

1. $\left(n+\frac{1}{2}\right)\frac{\pi }{3}$

2. $\left(n+\frac{1}{2}\right)\pi$

3. $\left(2n\pm \frac{1}{2}\right)\frac{\pi }{3}$

4. None of these

Q.

$(cosec\theta -cot\theta {)}^2$ = ?

(a) $\frac{1+cos\theta }{1-cos\theta }$ (b) $\frac{1-cos\theta }{1+cos\theta }$ (c) $\frac{1+sin\theta }{1-sin\theta }$ (d) none of these

Q.

If $5\tan \theta =4,$ then $\frac{(5\sin \theta -3\cos \theta )}{(5\sin \theta +2\cos \theta )}$ is

Q.

If (cos $\theta$ + sec $\theta$) = $\frac{5}{2}$ then $(co{s}^2\theta +se{c}^2\theta )$ = ?

(a) $\frac{21}{4}$ (b) $\frac{17}{4}$ (c) $\frac{29}{4}$ (d) $\frac{33}{4}$

Q. If $\Delta ABC$ is right angled at B, what is the value of $sin(A+C)$

Q.

What is $\cot \left(\frac{\theta }{2}\right)$ in terms of trigonometric functions?

Q.

How do you prove $se{c}^{2}x-{\tan }^{2}x=1$?

Q.

If A = ${30}^{\circ },$ verify that:

(i) $sin2A=\frac{2tanA}{1+ta{n}^{2}A}$ (ii) $cos2A=\frac{1-ta{n}^{2}A}{1+ta{n}^{2}A}$ (iii) $tan2A=\frac{2tanA}{1-ta{n}^{2}A}$

Q.

If 3 cot $\theta =2$, show that $\frac{4sin\theta -3cos\theta }{(2sin\theta +6cos\theta )}=\frac{1}{3}$.

Q.

If $(tan\theta +sin\theta )=m$ and $(tan\theta -sin\theta )=n$, prove that

$(m^2-n^2{)}^2=16mn.$

Q.

If $\sin \left(\theta \right)=\frac{24}{25}$ and $\left(\theta \right)$ lies in the second quadrant , then $\sec \left(\theta \right)+\tan \left(\theta \right)=$

1. $-3$

2. $-5$

3. $-7$

4. $-9$

Q.

If $3x=cosec\theta$ and $\frac{3}{x}=cot\theta$ then $3(x^2-\frac{1}{x^2})$ = ?

(a) $\frac{1}{27}$ (b) $\frac{1}{81}$ (c) $\frac{1}{3}$ (d) $\frac{1}{9}$

Q.

Prove- sin theta/cot theta + cosec theta=2+sin theta/cot theta- cosec theta

Q.

If ${\tan }^2\theta =1-e^2$, then $sec\theta +{\tan }^3\theta \cos ec\theta$ is equal to

1. $\sqrt{\left(2-e^2\right)}$

2. ${\left(2-e^2\right)}^{\frac{3}{2}}$

3. $\left(2-e^2\right)$

4. None of these

Q.

Prove that:

SinA/cotA+cosecA =2+(sinA/cotA - cosecA)

Q.

If sec $\theta =\frac{13}{5}$, show that $\frac{2sin\theta -3cos\theta }{4sin\theta -9cos\theta }=3$

Q.

If sec $\theta =\frac{5}{4}$, find the value of $\frac{sin\theta -2cos\theta }{tan\theta -cot\theta }$

Q.

The value of $\frac{sin{18}^{\circ }}{cos{72}^{\circ }}$ is ______.

1. 1

2. 0

3. -1

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

Q.

If $arg\left(z\right)=\theta$ then $arg\left(\overline{z}\right)$ equal to

Q.

If $\tan \left(\mathrm{\alpha }\right)+\cot \left(\mathrm{\alpha }\right)=2$ then find the value of ${\cot }^{20}\left(\mathrm{\alpha }\right)+{\tan }^{20}\left(\mathrm{\alpha }\right)$ is

1. $0$

2. $2$

3. $20$

4. $220$

Q.

If $sec\theta$ + $tan\theta$ = x, then $tan\theta$ is:

1. $\frac{(x^2+1)}{x}$

2. $\frac{(x^2+1)}{2x}$

3. $\frac{(x^2-1)}{2x}$

4. $\frac{(x^2-1)}{x}$