Trigonometric Identities
Trigonometric identities are equations involving trigonometric ratios that are true for all values of the angle for which the ratios are defined. They are covered in Chapter 8 (Introduction to Trigonometry) of the NCERT Class 10 textbook.
The three fundamental Pythagorean identities form the backbone of trigonometric simplifications and proofs:
- sin^2 theta + cos^2 theta = 1
- 1 + tan^2 theta = sec^2 theta
- 1 + cot^2 theta = cosec^2 theta
These identities are used extensively in proving other identities, simplifying expressions, and solving trigonometric equations. Mastery of these identities is essential for CBSE board exams.
The first identity — sin^2 theta + cos^2 theta = 1 — is derived directly from the Pythagoras Theorem applied to a right triangle. This connection between trigonometry and geometry makes it one of the most natural results in mathematics. The other two identities are derived from the first by simple algebraic division.
Understanding these identities is not just about memorising three equations. The key skill is recognising which form of the identity to apply in a given problem. For example, when you see sin^2 theta in an expression, you should immediately know that it can be replaced with 1 - cos^2 theta (or vice versa). Similarly, sec^2 theta - 1 should be recognised as tan^2 theta without conscious effort. This pattern recognition comes from extensive practice.
These identities are also the foundation for more advanced trigonometric work in Class 11 and 12, including compound angle formulas, multiple angle formulas, and the resolution of trigonometric equations. Mastering them in Class 10 creates a strong foundation for higher mathematics.
What is Trigonometric Identities - Proofs, Applications & Solved Examples?
Definition: A trigonometric identity is an equation involving trigonometric ratios that holds true for all permissible values of the variable (angle).
The three Pythagorean Identities:
Identity I: sin^2 theta + cos^2 theta = 1
Identity II: 1 + tan^2 theta = sec^2 theta
Identity III: 1 + cot^2 theta = cosec^2 theta
Rearranged forms:
- sin^2 theta = 1 - cos^2 theta
- cos^2 theta = 1 - sin^2 theta
- sec^2 theta - tan^2 theta = 1
- tan^2 theta = sec^2 theta - 1
- cosec^2 theta - cot^2 theta = 1
- cot^2 theta = cosec^2 theta - 1
Trigonometric Identities Formula
Trigonometric Identities — Complete Reference:
| Identity | Alternate Forms |
|---|---|
| sin^2 theta + cos^2 theta = 1 | sin^2 theta = 1 - cos^2 theta cos^2 theta = 1 - sin^2 theta |
| 1 + tan^2 theta = sec^2 theta | sec^2 theta - tan^2 theta = 1 tan^2 theta = sec^2 theta - 1 |
| 1 + cot^2 theta = cosec^2 theta | cosec^2 theta - cot^2 theta = 1 cot^2 theta = cosec^2 theta - 1 |
Useful factorisations:
- sec^2 theta - tan^2 theta = (sec theta - tan theta)(sec theta + tan theta) = 1
- cosec^2 theta - cot^2 theta = (cosec theta - cot theta)(cosec theta + cot theta) = 1
- sin^2 theta - cos^2 theta = (sin theta - cos theta)(sin theta + cos theta)
Derivation and Proof
Proof of the Three Identities:
Identity I: sin^2 theta + cos^2 theta = 1
- In a right triangle with angle theta: sin theta = p/h, cos theta = b/h (where p = opposite, b = adjacent, h = hypotenuse).
- sin^2 theta + cos^2 theta = p^2/h^2 + b^2/h^2 = (p^2 + b^2)/h^2
- By Pythagoras Theorem: p^2 + b^2 = h^2
- Therefore: sin^2 theta + cos^2 theta = h^2/h^2 = 1
Identity II: 1 + tan^2 theta = sec^2 theta
- Start with: sin^2 theta + cos^2 theta = 1
- Divide both sides by cos^2 theta:
- sin^2 theta/cos^2 theta + cos^2 theta/cos^2 theta = 1/cos^2 theta
- tan^2 theta + 1 = sec^2 theta
- Therefore: 1 + tan^2 theta = sec^2 theta
Identity III: 1 + cot^2 theta = cosec^2 theta
- Start with: sin^2 theta + cos^2 theta = 1
- Divide both sides by sin^2 theta:
- sin^2 theta/sin^2 theta + cos^2 theta/sin^2 theta = 1/sin^2 theta
- 1 + cot^2 theta = cosec^2 theta
- Therefore: 1 + cot^2 theta = cosec^2 theta
Types and Properties
Types of problems on trigonometric identities:
- Prove LHS = RHS: Start from one side, apply identities to reach the other side.
- Simplify expressions: Use identities to reduce complex expressions to simple forms.
- Evaluate given a ratio: Given sin theta = 3/5, find sec theta + tan theta (using identities, not triangle).
- Show that an expression = constant: Simplify to show the expression equals 0, 1, or another constant.
- Factorisation problems: Factor using a^2 - b^2 = (a-b)(a+b) with trig functions.
Difficulty levels in board exams:
- Easy (2 marks): Direct application of one identity. Example: Simplify (1 - cos^2 A) cosec^2 A.
- Medium (3 marks): Combining two identities or algebraic manipulation. Example: Prove cos theta/(1-tan theta) + sin theta/(1-cot theta) = sin theta + cos theta.
- Hard (4-5 marks): Multi-step proofs involving conjugates, factorisations, or higher powers. Example: Prove sin^6 A + cos^6 A = 1 - 3 sin^2 A cos^2 A.
Methods
Method 1: Start from the More Complex Side
Identify which side (LHS or RHS) is more complex. Simplify it step by step using identities until it matches the other side.
Method 2: Convert Everything to sin and cos
Replace tan, cot, sec, cosec with their expressions in terms of sin and cos. Then simplify.
Method 3: Use Algebraic Identities
- a^2 - b^2 = (a-b)(a+b)
- (a+b)^2 = a^2 + 2ab + b^2
- (a-b)^2 = a^2 - 2ab + b^2
Method 4: Multiply by Conjugate
Multiply numerator and denominator by the conjugate to simplify fractions involving (1 + sin theta), (sec theta - tan theta), etc.
Tips:
- NEVER move terms from one side to the other (this is not an equation to solve).
- Work on ONE side only and simplify to match the other.
- If stuck, try working from the other side.
- Look for sec^2 - tan^2 = 1 or cosec^2 - cot^2 = 1 patterns.
Strategy for exam success:
- Memorise the three identities and their rearranged forms. There are only 6-8 forms total.
- Practice at least 20 identity proofs from NCERT exercises and previous year papers.
- Classify each problem by type (direct, fraction, conjugate, cube) before starting.
- Time yourself: a 3-mark identity proof should take at most 5-6 minutes in an exam.
Writing the proof in exams:
- Write "LHS =" at the top and work downward.
- Show every algebraic step — do not skip steps.
- When using an identity, mention it: "(using sin^2 A + cos^2 A = 1)"
- End with "= RHS" and write "Hence proved" or "LHS = RHS. Proved."
Solved Examples
Example 1: Proving sin^2 theta + cos^2 theta = 1
Problem: Prove that sin^2 theta + cos^2 theta = 1.
Proof:
In a right triangle with hypotenuse h, opposite side p, adjacent side b:
- sin theta = p/h, cos theta = b/h
- sin^2 theta + cos^2 theta = p^2/h^2 + b^2/h^2 = (p^2 + b^2)/h^2
- By Pythagoras: p^2 + b^2 = h^2
- = h^2/h^2 = 1
Hence proved.
Example 2: Proving (1 - cos^2 theta)(1 + cot^2 theta) = 1
Problem: Prove that (1 - cos^2 theta)(1 + cot^2 theta) = 1.
LHS:
- 1 - cos^2 theta = sin^2 theta (Identity I)
- 1 + cot^2 theta = cosec^2 theta (Identity III)
- LHS = sin^2 theta x cosec^2 theta
- = sin^2 theta x (1/sin^2 theta)
- = 1 = RHS
Hence proved.
Example 3: Simplifying sec^2 theta - tan^2 theta
Problem: Simplify: (sec theta + tan theta)(sec theta - tan theta).
Solution:
- = sec^2 theta - tan^2 theta (difference of squares)
- = 1 (Identity II)
Answer: 1.
Example 4: Proving a Fraction Equals sin theta
Problem: Prove: (1 - cos^2 theta) x cosec theta = sin theta.
LHS:
- 1 - cos^2 theta = sin^2 theta
- LHS = sin^2 theta x cosec theta = sin^2 theta x (1/sin theta) = sin theta = RHS
Hence proved.
Example 5: Proving cos theta / (1 - tan theta) + sin theta / (1 - cot theta) = sin theta + cos theta
Problem: Prove: cos theta/(1 - tan theta) + sin theta/(1 - cot theta) = sin theta + cos theta.
LHS:
- tan theta = sin theta/cos theta, so 1 - tan theta = (cos theta - sin theta)/cos theta
- First term: cos theta / [(cos theta - sin theta)/cos theta] = cos^2 theta / (cos theta - sin theta)
- cot theta = cos theta/sin theta, so 1 - cot theta = (sin theta - cos theta)/sin theta
- Second term: sin theta / [(sin theta - cos theta)/sin theta] = sin^2 theta / (sin theta - cos theta)
- Note: sin theta - cos theta = -(cos theta - sin theta)
- Second term = sin^2 theta / [-(cos theta - sin theta)] = -sin^2 theta / (cos theta - sin theta)
- LHS = [cos^2 theta - sin^2 theta] / (cos theta - sin theta)
- = (cos theta - sin theta)(cos theta + sin theta) / (cos theta - sin theta)
- = cos theta + sin theta = RHS
Hence proved.
Example 6: Given sin theta, Find sec theta + tan theta
Problem: If sin theta = 3/5, find sec theta + tan theta.
Solution:
- cos^2 theta = 1 - sin^2 theta = 1 - 9/25 = 16/25, cos theta = 4/5
- sec theta = 5/4, tan theta = sin theta/cos theta = (3/5)/(4/5) = 3/4
- sec theta + tan theta = 5/4 + 3/4 = 8/4 = 2
Answer: 2.
Example 7: Proving (tan theta + sec theta - 1)/(tan theta - sec theta + 1) = (1 + sin theta)/cos theta
Problem: Prove the given identity.
LHS:
- Numerator: tan theta + sec theta - 1. Replace 1 with sec^2 theta - tan^2 theta:
- = tan theta + sec theta - (sec^2 theta - tan^2 theta)
- = tan theta + sec theta - (sec theta - tan theta)(sec theta + tan theta)
- = (sec theta + tan theta)[1 - (sec theta - tan theta)]
- = (sec theta + tan theta)(1 - sec theta + tan theta)
- Denominator: tan theta - sec theta + 1 = 1 - sec theta + tan theta
- LHS = (sec theta + tan theta)(1 - sec theta + tan theta) / (1 - sec theta + tan theta)
- = sec theta + tan theta
- = 1/cos theta + sin theta/cos theta = (1 + sin theta)/cos theta = RHS
Hence proved.
Example 8: Simplifying Using Conjugate
Problem: Simplify: (1 + sin theta) / cos theta.
Solution:
Multiply numerator and denominator by (1 - sin theta):
- = (1 + sin theta)(1 - sin theta) / [cos theta(1 - sin theta)]
- = (1 - sin^2 theta) / [cos theta(1 - sin theta)]
- = cos^2 theta / [cos theta(1 - sin theta)]
- = cos theta / (1 - sin theta)
So (1 + sin theta)/cos theta = cos theta/(1 - sin theta). Both equal sec theta + tan theta.
Example 9: Proving cosec theta - cot theta = 1/(cosec theta + cot theta)
Problem: Prove the identity.
RHS:
- 1/(cosec theta + cot theta)
- Multiply numerator and denominator by (cosec theta - cot theta):
- = (cosec theta - cot theta) / [(cosec theta + cot theta)(cosec theta - cot theta)]
- = (cosec theta - cot theta) / (cosec^2 theta - cot^2 theta)
- = (cosec theta - cot theta) / 1 (Identity III)
- = cosec theta - cot theta = LHS
Hence proved.
Example 10: Proving sqrt((1 - cos theta)/(1 + cos theta)) = cosec theta - cot theta
Problem: Prove the identity.
LHS:
- Rationalise inside the root: multiply numerator and denominator by (1 - cos theta):
- = sqrt[(1 - cos theta)^2 / (1 - cos^2 theta)]
- = sqrt[(1 - cos theta)^2 / sin^2 theta]
- = (1 - cos theta) / sin theta
- = 1/sin theta - cos theta/sin theta
- = cosec theta - cot theta = RHS
Hence proved.
Example 11: Proving tan A + cot A = sec A cosec A
Problem: Prove that tan A + cot A = sec A cosec A.
LHS:
- = sin A/cos A + cos A/sin A
- = (sin^2 A + cos^2 A)/(sin A cos A)
- = 1/(sin A cos A)
- = (1/sin A) x (1/cos A)
- = cosec A x sec A
- = sec A cosec A = RHS
Hence proved.
Example 12: Proving cos^4 A - sin^4 A = 1 - 2 sin^2 A
Problem: Prove that cos^4 A - sin^4 A = 1 - 2 sin^2 A.
LHS:
- = (cos^2 A - sin^2 A)(cos^2 A + sin^2 A) (difference of squares)
- = (cos^2 A - sin^2 A)(1)
- = cos^2 A - sin^2 A
- = (1 - sin^2 A) - sin^2 A
- = 1 - 2 sin^2 A = RHS
Hence proved.
Real-World Applications
Applications of trigonometric identities:
- Simplification: Reducing complex trigonometric expressions to simpler forms for evaluation.
- Proving other identities: More complex identities are proved using the three fundamental Pythagorean identities.
- Solving equations: Trigonometric equations are simplified using identities before solving.
- Calculus: Integration and differentiation of trigonometric functions rely heavily on these identities.
- Physics: Simplifying expressions in wave mechanics, optics, and oscillation problems.
- Engineering: Signal processing, AC circuit analysis, and Fourier transforms use trigonometric identities.
Key Points to Remember
- sin^2 theta + cos^2 theta = 1 (fundamental identity, derived from Pythagoras Theorem).
- 1 + tan^2 theta = sec^2 theta (derived by dividing by cos^2 theta).
- 1 + cot^2 theta = cosec^2 theta (derived by dividing by sin^2 theta).
- sec^2 theta - tan^2 theta = 1 and cosec^2 theta - cot^2 theta = 1.
- These can be factorised: (sec + tan)(sec - tan) = 1.
- To prove identities: work on ONE side, simplify to match the other.
- Convert tan, cot, sec, cosec to sin and cos when stuck.
- Use algebraic identities: a^2 - b^2 = (a-b)(a+b), (a+b)^2 expansion, etc.
- Multiplying by conjugate is a powerful technique for fraction-based identities.
- NEVER cross-multiply or move terms across the = sign (you are proving, not solving).
Practice Problems
- Prove: (1 + cos theta)(1 - cos theta) = sin^2 theta.
- Prove: sec^4 theta - sec^2 theta = tan^4 theta + tan^2 theta.
- Prove: (sin theta + cosec theta)^2 + (cos theta + sec theta)^2 = 7 + tan^2 theta + cot^2 theta.
- Simplify: (1 + tan^2 theta)(1 - sin theta)(1 + sin theta).
- If cos theta = 4/5, find (cosec theta - cot theta)/(cosec theta + cot theta).
- Prove: (tan theta - cot theta)^2 + 2 = sec^2 theta + cosec^2 theta.
- Prove: (sin theta/(1 - cos theta)) + ((1 - cos theta)/sin theta) = 2 cosec theta.
- Prove: sqrt((sec theta - 1)/(sec theta + 1)) + sqrt((sec theta + 1)/(sec theta - 1)) = 2 cosec theta.
Frequently Asked Questions
Q1. What are the three fundamental trigonometric identities?
sin^2 theta + cos^2 theta = 1, 1 + tan^2 theta = sec^2 theta, and 1 + cot^2 theta = cosec^2 theta.
Q2. How is the first identity proved?
Using Pythagoras Theorem in a right triangle: sin^2 theta + cos^2 theta = (p^2 + b^2)/h^2 = h^2/h^2 = 1.
Q3. How are Identities II and III derived?
By dividing Identity I by cos^2 theta (to get Identity II) and by sin^2 theta (to get Identity III).
Q4. What is the best approach to prove a trig identity?
Start from the more complex side. Convert to sin and cos if needed. Use algebraic identities (factorisation, conjugate multiplication). Simplify step by step to reach the other side.
Q5. Can you move terms across the equals sign when proving identities?
No. In a proof, you must work on one side and simplify it to match the other. Moving terms across treats it as an equation, which assumes what you are trying to prove.
Q6. What is the conjugate technique?
Multiply numerator and denominator by the conjugate (e.g., if you have 1/(1 + sin theta), multiply by (1 - sin theta)/(1 - sin theta)) to use the identity 1 - sin^2 theta = cos^2 theta.
Q7. Why is (sec theta + tan theta)(sec theta - tan theta) = 1?
Because sec^2 theta - tan^2 theta = 1 (Identity II), and (sec + tan)(sec - tan) = sec^2 - tan^2 by difference of squares.
Q8. Are trigonometric identity proofs important for board exams?
Yes. Identity proofs are among the most commonly asked questions in CBSE Class 10 board exams, typically worth 3-5 marks.
