KC Sinha Mathematics Solution Class 11 Chapter 5 Trigonometric Functions and Their graphs Exercise 5.1

 Exercise 5.1

Page no -5.11

Type 1

Question 1

1. $\sec ^{4} \theta-\sec ^{2} \theta=\tan ^{4} \theta+\tan ^{2} \theta$

2. $\frac{\cos \theta}{1+\sin \theta}+\frac{\cos \theta}{1-\sin \theta}=2 \sec \theta$

3. $\sin ^{6} \theta+\cos ^{6} \theta=1-3 \sin ^{2} \theta \cdot \cos ^{2} \theta$

4. $\sqrt{\frac{1+\cos \theta}{1-\cos \theta}}=\operatorname{cosec} \theta+\cot \theta$

5. $(1-\sin A-\cos A)^{2}=2(1-\sin A)(1-\cos A)$

6. $\tan A+\cot A=\sqrt{\sec ^{2} A+\operatorname{cosec}^{2} A}$

7. $\sin A(1+\tan A)+\cos A(1+\cot A)=\sec A+\operatorname{cosec} A$

8. $\frac{1}{\operatorname{cosec} \theta-\cot \theta}-\frac{1}{\sin \theta}=\frac{1}{\sin \theta}-\frac{1}{\operatorname{cosec} \theta+\cot \theta}$

9. $\frac{\tan \theta}{\sec \theta-1}+\frac{\tan \theta}{\sec \theta+1}=2 \operatorname{cosec} \theta$

10. $\cos \theta(\tan \theta+2)(2 \tan \theta+1)=2 \sec \theta+5 \sin \theta$

Page no 5.12

11. $(1+\tan \alpha \cdot \tan \beta)^{2}+(\tan \alpha-\tan \beta)^{2}=\sec ^{2} \alpha \cdot \sec ^{2} \beta$

12. $\frac{\cos ^{2} \alpha-\cos ^{2} \beta}{\cos ^{2} \alpha \cdot \cos ^{2} \beta}=\tan ^{2} \beta-\tan ^{2} \alpha$

13. $\frac{2 \sin \theta \cdot \cos \theta-\cos \theta}{1-\sin \theta+\sin ^{2} \theta-\cos ^{2} \theta}=\cot \theta$

Type 2

Question 14

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

Question 15

If $\sin ^{4} A+\sin ^{2} A=1$, prove that :

(i) $\frac{1}{\tan ^{4} A}+\frac{1}{\tan ^{2} A}=1$

(ii) $\tan ^{4} A-\tan ^{2} A=1$

Question 16

If $\tan \theta=\frac{a}{b}$, show that $\frac{a \sin \theta-b \cos \theta}{a \sin \theta+b \cos \theta}=\frac{a^{2}-b^{2}}{a^{2}+b^{2}}$

Question 17

If $\tan ^{2} \theta=1-k^{2}$, show that $\sec \theta+\tan ^{3} \theta \cdot \operatorname{cosec} \theta=\left(2-k^{2}\right)^{3 / 2}$

Question 18

If $\cos ^{2} B-\sin ^{2} B=\tan ^{2} A$, prove that :

$2 \cos ^{2} A-1=\cos ^{2} A-\sin ^{2} A=\tan ^{2} B .$

Question 19

If $T_{n}=\sin ^{n} \theta+\cos ^{n} \theta$, prove that $: 2 T_{6}-3 T_{4}+1=0$

Type -3

Question 20

If $\theta$ is an acute angle and $\sin \theta=\frac{a^{2}-b^{2}}{a^{2}+b^{2}},(a, b>0)$ find the values of $\tan \theta, \sec \theta$ and $\operatorname{cosec} \theta$

Question 21

 If $\sec A-\tan A=p, p>0$, find $\tan A$, $\sec A$ and $\sin A$ if $A$ is acute angle.

Question 22

If $A$ and $B$ are acute angles and $\frac{\sin A}{\sin B}=\sqrt{2}$ and $\frac{\tan A}{\tan B}=\sqrt{3}$, find $A$ and $B$.

Question 23

If $\tan \theta+\cot \theta=2$, find $\sin \theta$

Question 24

If $\sin x+\cos x=a$, then prove that :

$\sin ^{6} x+\cos ^{6} x=1-\frac{3}{4}\left(a^{2}-1\right)^{2} \text {, where } a^{2} \leq 2$

Type -3

Question 25

Eliminate $\theta$ from the following equations :

(i) $x=h+a \cos \theta, y=k+b \sin \theta$

(ii) $x \cos \theta-y \sin \theta=a, x \sin \theta+y \cos \theta=b$

Question 26

If $\sin x+\cos x=m$ and $\sec x+\operatorname{cosec} x=n$, prove that $n\left(m^{2}-1\right)=2 m$

Question 27

If $x=r \sin \theta, \cos \phi, y=$ $r \sin \theta \cdot \sin \phi, z=r \cos \theta$, show that :

$x^{2}+y^{2}+z^{2}=r^{2}$

Question 28

If $x \sin ^{3} \theta+y \cos ^{3} \theta=\sin \theta \cdot \cos \theta$ and $x \sin \theta-y \cos \theta=0$, prove that

$x^{2}+y^{2}=1 \text{ .} $