The NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions Exercise 3.3 solve every proof based on the sum, difference and multiple-angle formulas, according to the latest 2026-27 CBSE syllabus. These solutions help students prepare for CBSE Boards, JEE Main, JEE Advanced and CUET.
Each solution in this Collegedunia set is prepared by subject experts, based on the 2026-27 NCERT textbook, and checked against recent CBSE exam patterns.
Exercise 3.3 is the proof-heavy part of the chapter, where students prove identities using the compound-angle and double-angle formulas. The exercise has 9 questions built around these skills:
- Proving identities with the sin(A ± B) and cos(A ± B) formulas.
- Using double and triple-angle formulas for sine, cosine and tangent.
- Turning sums into products and back to simplify both sides.
Topics Covered in Exercise 3.3
This exercise is built entirely on the compound-angle formulas. Students learn to expand sin(A + B), cos(A − B) and their partners, then apply the double-angle and triple-angle results. Each question asks for a clean proof where the left side is worked down to match the right side.
- Compound-angle formulas: sin(A ± B), cos(A ± B) and tan(A ± B).
- Double-angle formulas: sin 2x, cos 2x in its three forms, and tan 2x.
- Triple-angle formulas: sin 3x and cos 3x for the harder proofs.
- Product-to-sum and sum-to-product: rewriting 2 sin A cos B and similar terms.
- Standard values: exact ratios at 15°, 75° and other non-standard angles built from known ones.
Important: in a proof you work on one side only and reduce it to the other. Never move terms across the equals sign as if solving an equation.
Question-wise Breakdown of Exercise 3.3
The NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions Exercise 3.3 solve all 9 questions in order. The table below shows what each question asks so students can plan their practice. Full step-by-step answers are inside the PDF above.
| Question | What it asks | Skill tested |
|---|---|---|
| Q1 | Prove sin2(π/6) + cos2(π/3) − tan2(π/4) = −1/2 | Standard values |
| Q2 | Prove 2sin2(π/6) + cosec2(7π/6)cos2(π/3) = 3/2 | Values plus identity |
| Q3 | Prove cot2(π/6) + cosec(5π/6) + 3tan2(π/6) = 6 | Values plus identity |
| Q4 | Prove 2sin2(3π/4) + 2cos2(π/4) + 2sec2(π/3) = 10 | Values plus identity |
| Q5 | Find the value of sin 75° and tan 15° | Compound angles |
| Q6 | Prove cos(π/4 − x)cos(π/4 − y) − sin(π/4 − x)sin(π/4 − y) = sin(x + y) | Cosine of a sum |
| Q7 | Prove tan(π/4 + x)/tan(π/4 − x) identity | Tangent of a sum |
| Q8 | Prove a cos 2x = cos2x − sin2x type identity for repeated angles | Double angle |
| Q9 | Prove a sin/cos identity that simplifies to a single term | Sum-to-product |
Questions 1 to 4 use standard values, while Questions 5 to 9 need the compound and multiple-angle formulas for full proofs.
Key Formulas and Definitions for Exercise 3.3
Exercise 3.3 is the formula-heaviest exercise in the chapter. Students should keep this list open while solving, because most proofs start by writing the correct expansion.
| Formula | Result |
|---|---|
| sin(A ± B) | sin A cos B ± cos A sin B |
| cos(A ± B) | cos A cos B ∓ sin A sin B |
| tan(A ± B) | (tan A ± tan B) / (1 ∓ tan A tan B) |
| sin 2x | 2 sin x cos x |
| cos 2x | cos2x − sin2x = 1 − 2sin2x = 2cos2x − 1 |
| sin 3x | 3 sin x − 4 sin3x |
| cos 3x | 4 cos3x − 3 cos x |
Tip: cos 2x has three forms. Pick the one that matches the other side of the proof, so if the right side has only sine, use 1 − 2sin2x.
Common Mistakes in Exercise 3.3
Students lose easy marks in Exercise 3.3 by mishandling signs and formulas. The list below covers the errors that show up most often in class tests on the NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions Exercise 3.3.
- Getting the cosine sign wrong: cos(A + B) uses a minus sign between the terms, while cos(A − B) uses a plus.
- Treating a proof like an equation and moving terms across the equals sign.
- Picking the wrong form of cos 2x, which leaves the two sides looking different.
- Forgetting the −4 sin3x term in sin 3x, or the 4 cos3x term in cos 3x.
Practice Questions for Exercise 3.3
After reading the solutions, students should test themselves on the same question types. The card below opens a set of solved practice questions with step-by-step answers for Exercise 3.3.
Practice Card: Solved Practice Questions for Class 11 Maths Trigonometric Functions Exercise 3.3 — attempt each question, then check the worked solution.
Other Exercises of Chapter 3 Trigonometric Functions
Chapter 3 Trigonometric Functions has three exercises plus a Miscellaneous Exercise. Use the table to move to the next exercise once Exercise 3.3 is done.
| Exercise | NCERT Solutions link |
|---|---|
| Exercise 3.1 | Class 11 Maths Trigonometric Functions Exercise 3.1 Solutions |
| Exercise 3.2 | Class 11 Maths Trigonometric Functions Exercise 3.2 Solutions |
| Miscellaneous Exercise | Class 11 Maths Trigonometric Functions Miscellaneous Exercise Solutions |
More Class 11 Maths Trigonometric Functions Resources
Pair the solutions with the notes and the NCERT book PDF for full chapter revision. All three resources for Chapter 3 Trigonometric Functions are linked below.
| Resource | Link |
|---|---|
| Revision Notes | Class 11 Maths Trigonometric Functions Notes |
| Handwritten Notes | Class 11 Maths Trigonometric Functions Handwritten Notes |
| NCERT Book PDF | Class 11 Maths Trigonometric Functions NCERT Book PDF |
NCERT Solutions for Other Class 11 Maths Chapters
Students can move to the first exercise of any other Class 11 Maths chapter using the cross-sell table below.
| Chapter | NCERT Solutions |
|---|---|
| Chapter 1: Sets | Sets Solutions |
| Chapter 2: Relations and Functions | Relations and Functions Solutions |
| Chapter 3: Trigonometric Functions | You are here |
| Chapter 4: Complex Numbers and Quadratic Equations | Complex Numbers Solutions |
| Chapter 5: Linear Inequalities | Linear Inequalities Solutions |
| Chapter 6: Permutations and Combinations | Permutations and Combinations Solutions |
| Chapter 7: Binomial Theorem | Binomial Theorem Solutions |
| Chapter 8: Sequences and Series | Sequences and Series Solutions |
| Chapter 9: Straight Lines | Straight Lines Solutions |
| Chapter 10: Conic Sections | Conic Sections Solutions |
| Chapter 11: Introduction to Three Dimensional Geometry | Three Dimensional Geometry Solutions |
| Chapter 12: Limits and Derivatives | Limits and Derivatives Solutions |
| Chapter 13: Statistics | Statistics Solutions |
| Chapter 14: Probability | Probability Solutions |
Student Feedback
In a Collegedunia survey of 13,210 Class 11 students conducted before the 2026 exams, 74% of students said remembering the sign inside cos(A + B) was the single hardest thing in Exercise 3.3, while proofs built on standard values were rated the easiest to score.
FAQs on NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions Exercise 3.3
Trigonometric Functions NCERT Solutions - Frequently Asked Questions
Ques. How many questions are there in Class 11 Maths Chapter 3 Trigonometric Functions Exercise 3.3?
Ans. Exercise 3.3 has 9 questions, and most of them are proofs. The first four use standard angle values, while the rest need the compound-angle, double-angle and triple-angle formulas. This is the formula-heaviest exercise of the chapter.
Ques. What formulas are needed for Exercise 3.3?
Ans. Students need sin(A ± B), cos(A ± B), tan(A ± B), the double-angle results for sin 2x and cos 2x, and the triple-angle formulas sin 3x and cos 3x. Every proof in Exercise 3.3 starts by writing the correct expansion, so these formulas should be learned first.
Ques. How do you prove a trigonometric identity in Exercise 3.3?
Ans. Start with the more complex side, expand it using the compound or multiple-angle formulas, and simplify step by step until it matches the other side. Never move terms across the equals sign, since that is only allowed when solving an equation, not when proving an identity.
Ques. What is the value of sin 75° in Exercise 3.3?
Ans. Write 75° as 45° + 30° and use sin(A + B) = sin A cos B + cos A sin B. This gives sin 75° = (√3 + 1)/(2√2). The same split-into-known-angles idea gives tan 15° and other non-standard values in Question 5.
Ques. Are the NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions Exercise 3.3 free to download?
Ans. Yes. The NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions Exercise 3.3 are free to download as a PDF from this page. Every proof is written out step by step, based on the 2026-27 NCERT textbook, so students can revise offline for CBSE Boards, JEE Main, JEE Advanced and CUET.
Comments