Proving Trigonometric Identities
Proving trigonometric identities is a core skill in Class 10 Mathematics. Unlike solving equations, proving an identity means showing that the left-hand side (LHS) equals the right-hand side (RHS) for all values of the variable.
This topic is part of Chapter 8 (Introduction to Trigonometry) in the NCERT Class 10 textbook. It builds on the three Pythagorean identities and requires algebraic manipulation skills.
Board exam questions on identity proofs typically carry 3-5 marks and require clear, step-by-step working.
The art of proving identities lies in strategic simplification. Unlike solving equations (where you find the value of a variable), proving identities requires you to demonstrate that two expressions are always equal. This means you must work on one side and transform it step-by-step until it matches the other side, using only valid algebraic operations and known identities.
The most common mistake students make is treating the identity as an equation and moving terms from one side to the other. This is logically invalid because it assumes what you are trying to prove. The correct approach is to work on one side only (usually the more complex side) and simplify it to match the other.
There are several powerful techniques for proving identities: converting to sin and cos, using Pythagorean identities, factoring using algebraic identities (difference of squares, sum/difference of cubes), and multiplying by conjugates. Mastering all these techniques is essential, as different problems require different approaches.
What is Proving Trigonometric Identities - Techniques, Steps & Solved Examples?
What is a trigonometric identity?
An identity is an equation that is true for all permissible values of the variable. It is NOT an equation to solve — it is a statement to prove.
Proving an identity means:
- Starting from one side (usually the more complex one)
- Using known identities and algebraic steps
- Arriving at the other side
The three fundamental identities used in proofs:
- sin^2 theta + cos^2 theta = 1
- 1 + tan^2 theta = sec^2 theta
- 1 + cot^2 theta = cosec^2 theta
Proving Trigonometric Identities Formula
Toolkit for Proving Identities:
Trigonometric identities:
- sin^2 A + cos^2 A = 1
- sec^2 A - tan^2 A = 1
- cosec^2 A - cot^2 A = 1
Reciprocal relations:
- tan A = sin A/cos A, cot A = cos A/sin A
- sec A = 1/cos A, cosec A = 1/sin A
Algebraic identities:
- (a + b)^2 = a^2 + 2ab + b^2
- (a - b)^2 = a^2 - 2ab + b^2
- a^2 - b^2 = (a + b)(a - b)
- a^3 + b^3 = (a + b)(a^2 - ab + b^2)
- a^3 - b^3 = (a - b)(a^2 + ab + b^2)
Derivation and Proof
General Strategy for Proving Identities:
- Identify the more complex side — start working on that side.
- Convert to sin and cos — replace tan, cot, sec, cosec with their definitions.
- Simplify algebraically — combine fractions, factor, expand as needed.
- Apply Pythagorean identities — replace sin^2 + cos^2 with 1, etc.
- Reach the other side — the final expression should match the simpler side exactly.
Common techniques:
- Taking LCM: When adding fractions with different denominators.
- Multiplying by conjugate: For expressions like 1/(1 + sin theta), multiply by (1 - sin theta)/(1 - sin theta).
- Factoring: Use a^2 - b^2 = (a-b)(a+b) for sec^2 - tan^2, etc.
- Splitting terms: Rewrite 1 as sin^2 + cos^2, or sec^2 as 1 + tan^2.
Types and Properties
Common types of identity proofs in board exams:
| Type | Example Pattern | Key Technique |
|---|---|---|
| Direct simplification | (1 - cos^2 A) cosec^2 A = 1 | Replace 1 - cos^2 with sin^2 |
| Fraction addition | 1/(1+sinA) + 1/(1-sinA) = 2sec^2A | Take LCM, simplify |
| Conjugate multiplication | (1-sinA)/(1+sinA) = (secA - tanA)^2 | Multiply by conjugate |
| Square root expressions | sqrt((1-cosA)/(1+cosA)) = cosecA - cotA | Rationalise, simplify |
| Higher powers | sin^6A + cos^6A = 1 - 3sin^2A cos^2A | Use a^3 + b^3 factorisation |
Board exam frequency analysis (based on previous years):
- Most commonly asked: Fraction addition type (LCM and simplify) — appears almost every year.
- Second most common: Conjugate multiplication type — appears frequently.
- Third: Direct substitution type — appears as a 2-mark short question.
- Occasional: Higher power identities (cubes, fourth powers) — appears every 2-3 years.
Marking scheme in board exams:
- 1 mark for correct LHS identification and initial conversion.
- 1-2 marks for intermediate algebraic steps.
- 1 mark for arriving at RHS with clear conclusion.
- Partial marks are awarded for correct intermediate steps even if the final answer is not reached.
Methods
Step-by-step method for proving identities:
- Write LHS and RHS separately. Choose the more complex side to work on.
- Convert to sin and cos (if the expression has tan, sec, cosec, cot).
- Simplify step by step. Show each step clearly.
- Use known identities when you see patterns like sin^2 + cos^2, sec^2 - tan^2, etc.
- State = RHS at the end.
Do NOT:
- Move terms from LHS to RHS (this is proving, not solving).
- Start with LHS = RHS (this assumes what you want to prove).
- Skip steps — board exams require clear step-by-step working.
Advanced techniques:
Technique 5: Splitting Terms
Rewrite 1 as sin^2 theta + cos^2 theta when you need to create a sum or difference. Similarly, rewrite sec^2 theta as 1 + tan^2 theta to introduce tan terms.
Technique 6: Working from Both Sides
If neither side simplifies easily, simplify BOTH sides independently and show they equal the same intermediate expression. This is acceptable as long as you do not cross terms between sides.
Technique 7: Substitution Check (NOT a proof, but useful for verification)
Substitute a specific angle (like 45 degrees) into both sides. If the values differ, the identity is FALSE (useful for ruling out incorrect identities). If they match, the identity MIGHT be true (proceed with algebraic proof).
Common exam patterns:
- Identities involving 1 + sin theta or 1 - sin theta: multiply by conjugate.
- Identities involving sec theta + tan theta: use (sec + tan)(sec - tan) = 1.
- Identities with cubes: use a^3 + b^3 = (a+b)(a^2-ab+b^2).
- Identities with fourth powers: use a^4 - b^4 = (a^2+b^2)(a^2-b^2).
Solved Examples
Example 1: Direct Substitution Proof
Problem: Prove: (1 + cos theta)(1 - cos theta)(1 + cot^2 theta) = 1.
LHS:
- (1 + cos theta)(1 - cos theta) = 1 - cos^2 theta = sin^2 theta
- 1 + cot^2 theta = cosec^2 theta = 1/sin^2 theta
- LHS = sin^2 theta x 1/sin^2 theta = 1 = RHS
Hence proved.
Example 2: Fraction Addition Proof
Problem: Prove: sin theta/(1 - cos theta) + (1 - cos theta)/sin theta = 2 cosec theta.
LHS:
- LCM = sin theta(1 - cos theta)
- = [sin^2 theta + (1 - cos theta)^2] / [sin theta(1 - cos theta)]
- = [sin^2 theta + 1 - 2cos theta + cos^2 theta] / [sin theta(1 - cos theta)]
- = [(sin^2 theta + cos^2 theta) + 1 - 2cos theta] / [sin theta(1 - cos theta)]
- = [1 + 1 - 2cos theta] / [sin theta(1 - cos theta)]
- = [2 - 2cos theta] / [sin theta(1 - cos theta)]
- = 2(1 - cos theta) / [sin theta(1 - cos theta)]
- = 2/sin theta = 2 cosec theta = RHS
Hence proved.
Example 3: Conjugate Multiplication Proof
Problem: Prove: (sec theta - tan theta)^2 = (1 - sin theta)/(1 + sin theta).
RHS:
- Multiply numerator and denominator by (1 - sin theta):
- = (1 - sin theta)^2 / [(1 + sin theta)(1 - sin theta)]
- = (1 - sin theta)^2 / (1 - sin^2 theta)
- = (1 - sin theta)^2 / cos^2 theta
- = [(1 - sin theta)/cos theta]^2
- = [1/cos theta - sin theta/cos theta]^2
- = [sec theta - tan theta]^2 = LHS
Hence proved.
Example 4: Proving sec^4 A - sec^2 A = tan^4 A + tan^2 A
Problem: Prove the identity.
LHS:
- = sec^2 A(sec^2 A - 1)
- = sec^2 A x tan^2 A (since sec^2 A - 1 = tan^2 A)
RHS:
- = tan^2 A(tan^2 A + 1)
- = tan^2 A x sec^2 A (since 1 + tan^2 A = sec^2 A)
LHS = RHS = sec^2 A x tan^2 A.
Hence proved.
Example 5: Proving sin^6 A + cos^6 A = 1 - 3 sin^2 A cos^2 A
Problem: Prove the identity.
LHS:
- Use a^3 + b^3 = (a + b)(a^2 - ab + b^2) with a = sin^2 A, b = cos^2 A:
- = (sin^2 A + cos^2 A)(sin^4 A - sin^2 A cos^2 A + cos^4 A)
- = 1 x (sin^4 A - sin^2 A cos^2 A + cos^4 A)
- Now: sin^4 A + cos^4 A = (sin^2 A + cos^2 A)^2 - 2 sin^2 A cos^2 A = 1 - 2 sin^2 A cos^2 A
- LHS = (1 - 2 sin^2 A cos^2 A) - sin^2 A cos^2 A
- = 1 - 3 sin^2 A cos^2 A = RHS
Hence proved.
Example 6: Proving (cosec A - sin A)(sec A - cos A) = 1/(tan A + cot A)
Problem: Prove the identity.
LHS:
- cosec A - sin A = 1/sin A - sin A = (1 - sin^2 A)/sin A = cos^2 A/sin A
- sec A - cos A = 1/cos A - cos A = (1 - cos^2 A)/cos A = sin^2 A/cos A
- LHS = (cos^2 A/sin A)(sin^2 A/cos A) = sin A cos A
RHS:
- tan A + cot A = sin A/cos A + cos A/sin A = (sin^2 A + cos^2 A)/(sin A cos A) = 1/(sin A cos A)
- 1/(tan A + cot A) = sin A cos A
LHS = RHS = sin A cos A.
Hence proved.
Example 7: Proving (1 + tan^2 A)/(1 + cot^2 A) = tan^2 A
Problem: Prove the identity.
LHS:
- 1 + tan^2 A = sec^2 A
- 1 + cot^2 A = cosec^2 A
- LHS = sec^2 A / cosec^2 A
- = (1/cos^2 A) / (1/sin^2 A)
- = sin^2 A / cos^2 A
- = tan^2 A = RHS
Hence proved.
Example 8: Proving tan A/(sec A - 1) + tan A/(sec A + 1) = 2 cosec A
Problem: Prove the identity.
LHS:
- LCM = (sec A - 1)(sec A + 1) = sec^2 A - 1 = tan^2 A
- Numerator: tan A(sec A + 1) + tan A(sec A - 1)
- = tan A(sec A + 1 + sec A - 1)
- = tan A(2 sec A)
- = 2 tan A sec A
- LHS = 2 tan A sec A / tan^2 A = 2 sec A / tan A
- = 2(1/cos A) / (sin A/cos A) = 2/sin A = 2 cosec A = RHS
Hence proved.
Example 9: Proving (sin A + cos A)^2 + (sin A - cos A)^2 = 2
Problem: Prove the identity.
LHS:
- (sin A + cos A)^2 = sin^2 A + 2 sin A cos A + cos^2 A = 1 + 2 sin A cos A
- (sin A - cos A)^2 = sin^2 A - 2 sin A cos A + cos^2 A = 1 - 2 sin A cos A
- LHS = (1 + 2 sin A cos A) + (1 - 2 sin A cos A) = 2 = RHS
Hence proved.
Example 10: Proving cot A - cos A / cot A + cos A = cosec A - 1 / cosec A + 1
Problem: Prove: (cot A - cos A)/(cot A + cos A) = (cosec A - 1)/(cosec A + 1).
LHS:
- cot A = cos A/sin A
- cot A - cos A = cos A/sin A - cos A = cos A(1/sin A - 1) = cos A(1 - sin A)/sin A
- cot A + cos A = cos A/sin A + cos A = cos A(1/sin A + 1) = cos A(1 + sin A)/sin A
- LHS = [cos A(1 - sin A)/sin A] / [cos A(1 + sin A)/sin A]
- = (1 - sin A)/(1 + sin A)
RHS:
- cosec A - 1 = 1/sin A - 1 = (1 - sin A)/sin A
- cosec A + 1 = 1/sin A + 1 = (1 + sin A)/sin A
- RHS = (1 - sin A)/(1 + sin A)
LHS = RHS. Hence proved.
Example 11: Proving (1 + sin A)/(1 - sin A) = (sec A + tan A)^2
Problem: Prove the identity.
RHS:
- = (sec A + tan A)^2
- = (1/cos A + sin A/cos A)^2
- = ((1 + sin A)/cos A)^2
- = (1 + sin A)^2/cos^2 A
- = (1 + sin A)^2/(1 - sin^2 A)
- = (1 + sin A)^2/((1 - sin A)(1 + sin A))
- = (1 + sin A)/(1 - sin A) = LHS
Hence proved.
Example 12: Proving (sin A - cos A + 1)/(sin A + cos A - 1) = 1/(sec A - tan A)
Problem: Prove the identity.
LHS:
Divide numerator and denominator by cos A:
- = (tan A - 1 + sec A)/(tan A + 1 - sec A)
- = (sec A + tan A - 1)/(tan A - sec A + 1)
Replace 1 in numerator with sec^2 A - tan^2 A:
- = (sec A + tan A - (sec^2 A - tan^2 A))/(tan A - sec A + 1)
- = ((sec A + tan A)(1 - sec A + tan A))/(tan A - sec A + 1)
- = sec A + tan A
Now: 1/(sec A - tan A) = (sec A + tan A)/((sec A - tan A)(sec A + tan A)) = (sec A + tan A)/1 = sec A + tan A.
LHS = RHS = sec A + tan A. Hence proved.
Real-World Applications
Applications of proving identities:
- Simplifying expressions: The techniques used in proofs are directly applicable to simplifying complex trigonometric expressions in exams and real-world problems.
- Solving equations: Recognising identity patterns helps convert trigonometric equations into solvable forms.
- Higher mathematics: Identity-proving skills are foundational for calculus, complex analysis, and Fourier series.
- Physics and engineering: Simplifying wave equations, AC circuit analysis, and signal processing expressions requires these skills.
- Competitive exams: JEE, NEET, and other competitive exams extensively test identity-proving abilities.
Key Points to Remember
- An identity is true for ALL values; a proof must show LHS = RHS through valid algebraic steps.
- Start from the more complex side.
- Convert everything to sin and cos when stuck.
- Use the three Pythagorean identities: sin^2 + cos^2 = 1, 1 + tan^2 = sec^2, 1 + cot^2 = cosec^2.
- Use algebraic identities: a^2 - b^2, (a+b)^2, (a-b)^2, a^3 +/- b^3.
- Conjugate multiplication is key for expressions with (1 + sin theta), (sec theta - tan theta), etc.
- Never transpose terms from one side to the other.
- Never start by writing LHS = RHS (you cannot assume what you need to prove).
- Show each step clearly — marks are awarded for the working, not just the final answer.
- Practice a variety of identity types: direct, fraction, conjugate, higher-power, and mixed.
Practice Problems
- Prove: (sin theta + cos theta)^2 - 1 = 2 sin theta cos theta.
- Prove: (1 - sin theta)/(1 + sin theta) = (sec theta - tan theta)^2.
- Prove: (tan theta + cot theta)^2 = sec^2 theta + cosec^2 theta.
- Prove: cos^4 A - sin^4 A = cos^2 A - sin^2 A.
- Prove: (cosec theta + cot theta)/(cosec theta - cot theta) = (1 + cos theta)^2/sin^2 theta.
- Prove: tan^2 A/(1 + tan^2 A) + cot^2 A/(1 + cot^2 A) = 1.
- Prove: (sin A - 2 sin^3 A)/(2 cos^3 A - cos A) = tan A.
- Prove: (1 + sec A)/sec A = sin^2 A/(1 - cos A).
Frequently Asked Questions
Q1. What does it mean to prove a trigonometric identity?
It means showing that the LHS equals the RHS for all values of the variable, using known identities and algebraic manipulation. You work on one side and simplify it to match the other.
Q2. Which side should you start from?
Start from the more complex side. If both sides look equally complex, start from the LHS (left-hand side).
Q3. Can you work on both sides simultaneously?
It is acceptable to simplify both sides independently and show they equal the same expression. But the standard approach is to work on one side only.
Q4. What is the most common mistake in identity proofs?
Moving terms from one side to the other (transposing). This treats the identity as an equation, which assumes what you are trying to prove.
Q5. When should you use the conjugate technique?
When you see expressions like 1/(1 + sin A), (1 - cos A)/(1 + cos A), or sqrt((1-sin A)/(1+sin A)). Multiplying by the conjugate creates differences of squares that simplify using Pythagorean identities.
Q6. How many marks do identity proofs carry in board exams?
Typically 3-5 marks. Clear step-by-step working is essential. Each step may carry partial marks.
Q7. What if the identity seems impossible to prove?
Try: (1) Convert everything to sin and cos. (2) Take LCM for fractions. (3) Factor using algebraic identities. (4) Try working from the other side.
Q8. Is it necessary to memorise identities for the exam?
You must know the three Pythagorean identities, the reciprocal relations, and basic algebraic identities. With these, any Class 10 identity can be proved.
