Class 11 Maths Chapter 3 Trigonometric Functions – Exercise 3.2 NCERT Solutions

Introduction

Exercise 3.2 focuses on trigonometric identities and their applications. You will learn how to prove fundamental identities, simplify trigonometric expressions, and solve problems using these identities. Mastery of identities is essential for higher‑level trigonometry and calculus.

Key Identities

1. Pythagorean Identities:

$${\sin }^2\theta +{\cos }^2\theta =1$$

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

$$1+{\cot }^2\theta ={\csc }^2\theta$$

2. Quotient Identities:

$$\tan \theta =\frac{\sin \theta }{\cos \theta },\cot \theta =\frac{\cos \theta }{\sin \theta }$$

3. Reciprocal Identities:

$$\csc \theta =\frac{1}{\sin \theta },\sec \theta =\frac{1}{\cos \theta }$$

NCERT Questions with Solutions (10)

Q1. Prove that $\frac{1-{\sin }^2\theta }{{\cos }^2\theta }=1$.

$$\frac{1-{\sin }^2\theta }{{\cos }^2\theta }=\frac{{\cos }^2\theta }{{\cos }^2\theta }=1$$

Q2. Prove that ${\sec }^2\theta -{\tan }^2\theta =1$. From identity: $1+{\tan }^2\theta ={\sec }^2\theta$. Rearranging gives result.

Q3. Prove that ${\csc }^2\theta -{\cot }^2\theta =1$. From identity: $1+{\cot }^2\theta ={\csc }^2\theta$. Rearranging gives result.

Q4. Simplify $\frac{{\sin }^2\theta }{1+\cos \theta }$.

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

Q5. Simplify $\frac{1-\cos \theta }{\sin \theta }$.

$$\frac{1-\cos \theta }{\sin \theta }=\frac{2{\sin }^2\frac{\theta }{2}}{2\sin \frac{\theta }{2}\cos \frac{\theta }{2}}=\tan \frac{\theta }{2}$$

Q6. Prove that $\frac{\tan \theta }{1+\sec \theta }=\sin \theta$.

$$\frac{\tan \theta }{1+\sec \theta }=\frac{\sin \theta \mathrm{/}\cos \theta }{1+1\mathrm{/}\cos \theta }=\sin \theta$$

Q7. Prove that $\frac{1+\cos \theta }{\sin \theta }=\cot \frac{\theta }{2}$. Using half‑angle identities, result holds.

Q8. Prove that $\frac{1-\cos \theta }{\sin \theta }=\tan \frac{\theta }{2}$. Using half‑angle identities, result holds.

Q9. Simplify $\frac{\sin \theta }{1+\cos \theta }$.

$$\frac{\sin \theta }{1+\cos \theta }=\tan \frac{\theta }{2}$$

Q10. Simplify $\frac{\cos \theta }{1+\sin \theta }$.

$$\frac{\cos \theta }{1+\sin \theta }=\cot \frac{\theta }{2}$$

FAQs (10)

1. What are trigonometric identities? Equations true for all values of angle.

2. What is Pythagorean identity? ${\sin }^2\theta +{\cos }^2\theta =1$.

3. What is quotient identity? $\tan \theta =\frac{\sin \theta }{\cos \theta }$.

4. What is reciprocal identity? $\csc \theta =\frac{1}{\sin \theta }$.

5. What is half‑angle identity? Expresses functions in terms of half angle.

6. Why are identities important? Used to simplify trigonometric expressions.

7. Can identities be used in proofs? Yes, widely used in mathematics.

8. What is domain of trigonometric identities? All real values where functions are defined.

9. What is range of sine and cosine? $[-1,1]$.

10. Why is this exercise important? It builds foundation for solving trigonometric equations.

Conclusion

Exercise 3.2 has 10 solved questions and 10 FAQs that strengthen your understanding of trigonometric identities and simplifications. This builds the foundation for advanced trigonometric problem solving in Class 11 Maths.

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