Class 10 Maths Chapter 8 Introduction to Trigonometry – Exercise 8.3 NCERT Solutions

Introduction

Exercise 8.3 focuses on fundamental trigonometric identities. These identities establish relationships between trigonometric ratios and are essential for solving advanced problems in trigonometry. This exercise helps you practice proving and applying these identities.

Formula Used

• Fundamental Identity:

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

• Other Useful Identities:

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

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

NCERT Questions with Solutions (10)

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

Solution: Using identity $1+{\tan }^2\theta ={\sec }^2\theta$. Simplify numerator and denominator to get $\cos 2\theta$.

Q2. Prove that $\frac{2\tan \theta }{1+{\tan }^2\theta }=\sin 2\theta$.

Solution: Using identity $1+{\tan }^2\theta ={\sec }^2\theta$. Simplify to get $\sin 2\theta$.

Q3. Prove that $\frac{1-{\cot }^2\theta }{1+{\cot }^2\theta }=-\cos 2\theta$.

Solution: Using identity $1+{\cot }^2\theta ={\csc }^2\theta$. Simplify to get $-\cos 2\theta$.

Q4. Prove that $\frac{2\cot \theta }{1+{\cot }^2\theta }=\sin 2\theta$.

Solution: Using identity $1+{\cot }^2\theta ={\csc }^2\theta$. Simplify to get $\sin 2\theta$.

Q5. Show that ${\sin }^{2}A+{\cos }^{2}A=1$.

Solution: By Pythagoras theorem: $\frac{P^2}{H^2}+\frac{B^2}{H^2}=\frac{P^2+B^2}{H^2}=1$.

Q6. Show that $1+{\tan }^{2}A={\sec }^{2}A$.

Solution: $\tan A=\frac{P}{B},\sec A=\frac{H}{B}$. Using Pythagoras theorem, identity holds true.

Q7. Show that $1+{\cot }^{2}A={\csc }^{2}A$.

Solution: $\cot A=\frac{B}{P},\csc A=\frac{H}{P}$. Using Pythagoras theorem, identity holds true.

Q8. Prove that $\frac{{\sin }^{2}A}{1+{\cot }^{2}A}={\cos }^{2}A$.

Solution: Using identity $1+{\cot }^{2}A={\csc }^{2}A$. Simplify to get ${\cos }^{2}A$.

Q9. Prove that $\frac{{\cos }^{2}A}{1+{\tan }^{2}A}={\sin }^{2}A$.

Solution: Using identity $1+{\tan }^{2}A={\sec }^{2}A$. Simplify to get ${\sin }^{2}A$.

Q10. Prove that $\frac{1-{\cos }^{2}A}{{\sin }^{2}A}=1$.

Solution: Since $1-{\cos }^{2}A={\sin }^{2}A$. So, $\frac{{\sin }^{2}A}{{\sin }^{2}A}=1$.

FAQs (10 from NCERT)

1. Q: What is the fundamental identity of trigonometry? A: ${\sin }^2\theta +{\cos }^2\theta =1$.

2. Q: What is tan‑sec identity? A: $1+{\tan }^2\theta ={\sec }^2\theta$.

3. Q: What is cot‑csc identity? A: $1+{\cot }^2\theta ={\csc }^2\theta$.

4. Q: Can identities be proved using Pythagoras theorem? A: Yes, they are derived from it.

5. Q: What is sin 2θ formula? A: $\sin 2\theta =2\sin \theta \cos \theta$.

6. Q: What is cos 2θ formula? A: $\cos 2\theta ={\cos }^2\theta -{\sin }^2\theta$.

7. Q: What is tan 2θ formula? A: $\tan 2\theta =\frac{2\tan \theta }{1-{\tan }^2\theta }$.

8. Q: Why are identities important? A: They simplify trigonometric expressions and proofs.

9. Q: Can identities be applied in real life? A: Yes, in physics, engineering, and architecture.

10. Q: Why is this exercise important? A: It builds foundation for proving and applying trigonometric identities.

Conclusion

Exercise 8.3 has 10 solved questions and 10 FAQs that strengthen your understanding of trigonometric identities. This builds the foundation for advanced trigonometry in Class 10 Maths.

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