Trigonometric Identities

Q.

Question 1
Show that in a right angled triangle, the hypotenuse is the longest side.

Q.

In a $∆ABC,b=\surd 3,c=1\text{and}\angle A=30°,$ then the largest angle of the triangle is

1. $60°$

2. $135°$

3. $90°$

4. $120°$

Q. If 15 cot A = 8, find tan A.

1. $\frac{8}{14}$
2. $\frac{15}{8}$
3. $\frac{17}{8}$
4. $\frac{15}{17}$

Q.

Choose the correct option. Justify your choice. $9{\sec }^{2}A-9{\tan }^{2}A=$

1. $1$

2. $9$

3. $8$

4. $0$

Q.

A bridge across a river makes an angle of $ {45}^{0}$ with the river bank. If the length of the bridge across the river is $ 150 m$, what is the width of the river ?

Q.

If $cos\theta =\frac{4}{5}$, find all other trigonometric ratios of angle $\theta$

Q. $9se{c}^{2}A-9ta{n}^{2}A$ is equal to

1. 1
2. 9
3. 8
4. 0

Q.

A triangle ABC is right angled at B. Find the value of

$\frac{\sec A.\sin C-\tan A\tan C}{\sin B}$

Q. The value of $tan\theta$ in given $△$ ABC in terms of a, b and c is

1. $\frac{b}{c}$
2. $\frac{b}{a}$
3. $\frac{a}{c}$

Q.

If ${\cos }^6\alpha +{\sin }^6\alpha +K{\sin }^2\left(2\alpha \right)=1$, then $K=?$

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

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

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

4. $2$

Q.

Two airplanes leave an airport at the same time. The first flies $150\frac{km}{h}$ in a direction of $320°$. The second flies $200\frac{km}{h}$ in a direction of $200°$. After $3hr$ , how far apart are the planes?

Q.

Statement1: $cose{c}^{2}A$ - $co{t}^{2}A$ = 1

Statement2: 1 + $ta{n}^{2}A$ = $se{c}^{2}A$

1. Statement1 is true and Statement2 is false

2. Statement1 is false and Statement2 is true

3. Statement1 are Statement2 are true

4. Statement1 are Statement2 are false

Q. Show that:
$\left(i\right)\tan {10}^{\circ }\tan {15}^{\circ }\tan {75}^{\circ }\tan {80}^{\circ }=1$
$\left(ii\right)\sin {42}^{\circ }\sec {48}^{\circ }+\cos {42}^{\circ }$ cosec ${48}^{\circ }=2$

Q. If $cosecA+\cot A=m$, show that $\frac{m^2-1}{m^2+1}=\cos A$.

Q. ${\left(\sin A+\cos ecA\right)}^2+{\left(\cos A+\sec A\right)}^2={\tan }^{2}A+{\cot }^{2}A+7$

Q. $(1+sec\theta )(1-sec\theta )+ta{n}^2\theta =$

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

Q. $9{\sec }^{2}A-9{\tan }^{2}A=..........$

1. 9
2. 1
3. 0
4. 3

Q. If $\sin \theta +\cos \theta =\sqrt{3}$, then prove that $\tan \theta +\cot \theta =1$

Q. Find the value of: $\sin \left(\frac{1}{4}{\sin }^{-1}\frac{\sqrt{63}}{8}\right)$

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

Q. If $3\cot A=4$, check whether $\frac{1-{\tan }^{2}A}{1+{\tan }^{2}A}={\cos }^{2}A-{\sin }^{2}A$ or not

Q. If $ta{n}^{2}x=1-a^2$, prove that $secx+ta{n}^{3}x.cosecx=(2-a^2{)}^{\frac{3}{2}}$ . Also find the values of a for which the above result holds true .

Q.

$\frac{ta{n}^2\theta }{1+ta{n}^2\theta }$ + $\frac{co{t}^2\theta }{1+co{t}^2\theta }$ =

1. 0

2. -1

3. 1

4. 2

Q. In the given figure, find the value of $sec\frac{A}{2}+\cos ec\frac{A}{2}$

1. $\frac{20}{12}$
2. $\frac{35}{12}$
3. $\frac{12}{35}$
4. $\frac{37}{12}$

Q. Prove that $\frac{{cos}^3\theta +{sin}^3\theta }{cos\theta +sin\theta }+\frac{{cos}^3\theta -{sin}^3\theta }{cos\theta -sin\theta }=2$

Q.

The value of $tan{58}^{\circ }tan{32}^{\circ }tan{57}^{\circ }tan{33}^{\circ }$ is

1. 2

2. 3

3. 1

4. 0

Q. $\frac{1+{\tan }^{2}A}{1+{\cot }^{2}A}$ is equal to

1. 
2. 
3. 
4. 

Q. $2ta{n}^{-1}(cosecta{n}^{-1}x-tanco{t}^{-1}x)$ is equal to

1. $co{s}^{-1}x$
2. $ta{n}^{-1}x$
3. $si{n}^{-1}x$
4. $co{t}^{-1}x$

Q. If $\cos \theta -\sin \theta =\sqrt{2}\sin \theta$, prove that $\cos \theta +\sin \theta =\sqrt{2}\cos \theta$.

Q.

If $0^{\circ }<\theta <{90}^{\circ }$, then $\sqrt{se{c}^2\theta +cose{c}^2\theta }$ =

1. tan θ + cot θ

2. tan θ - cot θ

3. - tan θ + cot θ

4. - tan θ - cot θ

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