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{ .}
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