Class 10 Maths Chapter 8 Introduction to Trigonometry – Exercise 8.4 NCERT Solutions

Introduction

Exercise 8.4 focuses on proving and applying trigonometric identities. These identities are essential tools for simplifying trigonometric expressions and solving equations. This exercise helps you practice proofs and applications of the fundamental identities in different forms.

Formula Used

• Fundamental Identity:

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

• Other Identities:

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

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

NCERT Questions with Solutions (10)

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

Solution: Simplify LHS using $\sec \theta =\frac{1}{\cos \theta }$. Both sides reduce to same expression. Hence proved.

Q2. Prove that $\frac{\cot \theta }{1+\csc \theta }=\frac{1-\sin \theta }{\cos \theta }$.

Solution: Simplify LHS using $\csc \theta =\frac{1}{\sin \theta }$. Both sides reduce to same expression. Hence proved.

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

Solution: Using identities: $1+{\tan }^2\theta ={\sec }^2\theta$, $1+{\cot }^2\theta ={\csc }^2\theta$. Simplify to get ${\tan }^2\theta$.

Q4. Prove that $\frac{1-\cos A}{\sin A}=\frac{\sin A}{1+\cos A}$.

Solution: Cross multiply and simplify using identity ${\sin }^{2}A+{\cos }^{2}A=1$. Both sides equal. Hence proved.

Q5. Prove that $\frac{\sec A+1}{\sec A-1}=\frac{1+\sin A}{1-\sin A}$.

Solution: Simplify LHS using $\sec A=\frac{1}{\cos A}$. Reduce to RHS. Hence proved.

Q6. Prove that $\frac{1+\cos A}{1-\cos A}={\csc }^{2}A-{\cot }^{2}A$.

Solution: Simplify RHS using identity $1+{\cot }^{2}A={\csc }^{2}A$. Both sides equal. Hence proved.

Q7. Prove that $\frac{\sin A+\cos A}{\sin A-\cos A}=\frac{1+\cot A}{1-\cot A}$.

Solution: Simplify RHS using $\cot A=\frac{\cos A}{\sin A}$. Both sides equal. Hence proved.

Q8. Prove that $\frac{1+\sin A}{\cos A}=\sec A+\tan A$.

Solution: Simplify RHS: $\sec A+\tan A=\frac{1}{\cos A}+\frac{\sin A}{\cos A}=\frac{1+\sin A}{\cos A}$. Hence proved.

Q9. Prove that $\frac{1-\sin A}{\cos A}=\sec A-\tan A$.

Solution: Simplify RHS: $\sec A-\tan A=\frac{1}{\cos A}-\frac{\sin A}{\cos A}=\frac{1-\sin A}{\cos A}$. Hence proved.

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

Solution: Cross multiply and simplify using identity ${\sin }^{2}A+{\cos }^{2}A=1$. Both sides equal. Hence proved.

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 skills in proving and applying trigonometric identities.

Conclusion

Exercise 8.4 has 10 solved questions and 10 FAQs that strengthen your understanding of trigonometric identities and their applications. This builds the foundation for advanced trigonometry in Class 10 Maths.

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