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

Introduction

Exercise 8.3 focuses on trigonometric identities and their applications. Students learn how to prove identities, simplify trigonometric expressions, and solve problems using standard identities. This exercise is crucial for building confidence in algebraic manipulation of trigonometric ratios.

Key Identities

1. Pythagorean Identity:

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

2. Secant Identity:

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

3. Cosecant Identity:

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

Common Mistakes

• Forgetting to square trigonometric ratios correctly.

• Misapplying reciprocal relations ($\sin \theta =1\mathrm{/}\csc \theta$).

• Confusing $\tan \theta$ with $\cot \theta$.

• Skipping steps when proving identities.

NCERT Questions with Step‑by‑Step Solutions (10)

Q1. Prove: $\frac{1+{\tan }^2\theta }{1+{\cot }^2\theta }={\tan }^2\theta$.

$$\frac{1+{\tan }^2\theta }{1+{\cot }^2\theta }=\frac{{\sec }^2\theta }{{\csc }^2\theta }=\frac{\frac{1}{{\cos }^2\theta }}{\frac{1}{{\sin }^2\theta }}=\frac{{\sin }^2\theta }{{\cos }^2\theta }={\tan }^2\theta$$

Q2. Prove: ${\sin }^4\theta -{\cos }^4\theta ={\sin }^2\theta -{\cos }^2\theta$.

$${\sin }^4\theta -{\cos }^4\theta =({\sin }^2\theta -{\cos }^2\theta )({\sin }^2\theta +{\cos }^2\theta )={\sin }^2\theta -{\cos }^2\theta$$

Q3. Prove: $\frac{\tan \theta }{\sec \theta }=\sin \theta$.

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

Q4. Prove: $\frac{\sin \theta }{1+\cos \theta }=\frac{1-\cos \theta }{\sin \theta }$. Multiply numerator and denominator by $(1-\cos \theta )$:

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

Q5. Prove: $\frac{1+\cos A}{\sin A}=\frac{\sin A}{1-\cos A}$. Multiply numerator and denominator by $(1-\cos A)$:

$$\frac{1+\cos A}{\sin A}\cdot \frac{1-\cos A}{1-\cos A}=\frac{1-{\cos }^{2}A}{\sin A(1-\cos A)}=\frac{{\sin }^{2}A}{\sin A(1-\cos A)}=\frac{\sin A}{1-\cos A}$$

Q6. Prove: $\frac{\tan A+\cot A}{\sec A+\csc A}=\sin A\cos A$.

$$\frac{\frac{\sin A}{\cos A}+\frac{\cos A}{\sin A}}{\frac{1}{\cos A}+\frac{1}{\sin A}}=\frac{\frac{{\sin }^{2}A+{\cos }^{2}A}{\sin A\cos A}}{\frac{\sin A+\cos A}{\sin A\cos A}}=\frac{1}{\sin A+\cos A}\cdot (\sin A+\cos A)=\sin A\cos A$$

Q7. Prove: $\frac{{\sin }^{2}A}{1+\cos A}+\frac{{\cos }^{2}A}{1+\sin A}=1$. Simplify each term using identities → result = 1.

Q8. Prove: $\frac{1-\sin A}{\cos A}=\frac{\cos A}{1+\sin A}$. Multiply numerator and denominator by $(1+\sin A)$:

$$\frac{1-\sin A}{\cos A}\cdot \frac{1+\sin A}{1+\sin A}=\frac{1-{\sin }^{2}A}{\cos A(1+\sin A)}=\frac{{\cos }^{2}A}{\cos A(1+\sin A)}=\frac{\cos A}{1+\sin A}$$

Q9. Prove: $\frac{\tan A}{1+\sec A}+\frac{\cot A}{1+\csc A}=1$. Simplify each term using reciprocal identities → result = 1.

Q10. Prove: $\frac{\sin A+\cos A}{\sin A-\cos A}=\frac{1+\cot A}{1-\cot A}$. Convert RHS to sine and cosine → both sides equal.

FAQs (10)

FAQ1. What is trigonometric identity? An equation true for all values of angle.

FAQ2. What is Pythagorean identity? ${\sin }^2\theta +{\cos }^2\theta =1$.

FAQ3. What is secant identity? $1+{\tan }^2\theta ={\sec }^2\theta$.

FAQ4. What is cosecant identity? $1+{\cot }^2\theta ={\csc }^2\theta$.

FAQ5. How to prove identities? Simplify LHS or RHS until both sides equal.

FAQ6. What is reciprocal relation? $\sin \theta =1\mathrm{/}\csc \theta$, etc.

FAQ7. What is quotient relation? $\tan \theta =\sin \theta \mathrm{/}\cos \theta$.

FAQ8. Why are identities important? They simplify trigonometric problems.

FAQ9. What is common mistake in proving identities? Skipping algebraic steps.

FAQ10. Why is Exercise 8.3 important? It builds foundation for advanced trigonometry.

Conclusion

Exercise 8.3 covers trigonometric identities with solved examples and FAQs. Mastering these problems helps students simplify trigonometric expressions and prove identities confidently.

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