Trigonometric Identities
Introduction
Trigonometric identities are equations involving trigonometric ratios that hold true for all values of the angle. In ICSE Class 10, you learn the three fundamental identities and use them to prove more complex identities.
Fundamental Identities
- sin²A + cos²A = 1
- sec²A = 1 + tan²A
- cosec²A = 1 + cot²A
Derived Forms
From identity (1):
- sin²A = 1 − cos²A
- cos²A = 1 − sin²A
From identity (2):
- tan²A = sec²A − 1
- sec²A − tan²A = 1
From identity (3):
- cot²A = cosec²A − 1
- cosec²A − cot²A = 1
ICSE-Specific Angle Values
Memorise these values for ICSE problems:
| Angle | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin | 0 | ¹/₂ | ¹/√2 | √3/2 | 1 |
| cos | 1 | √3/2 | ¹/√2 | ¹/₂ | 0 |
| tan | 0 | ¹/√3 | 1 | √3 | ∞ |
Proving Trigonometric Identities — Strategy
Step 1: Choose the more complex side
Start with the side that looks more complicated (usually LHS).
Step 2: Convert to sine and cosine
Express all ratios in terms of sin and cos if simplification is needed.
Step 3: Use algebraic manipulation
Factorise, rationalise, or use the fundamental identities.
Step 4: Simplify step-by-step
Work towards the simpler side. Never cross-multiply across the equality sign.
Step 5: Verify the final expression matches
The simplified form must exactly match the other side.
Worked Examples
Example 1: Basic Identity Proof
Prove that: (1 − cos²A) cosec²A = 1
Solution: LHS = (1 − cos²A) × cosec²A = sin²A × cosec²A (since sin²A + cos²A = 1) = sin²A × 1/sin²A = 1 = RHS ✓
Example 2: Using sec²A − tan²A = 1
Prove that: sec⁴A − sec²A = tan⁴A + tan²A
Solution: LHS = sec⁴A − sec²A = sec²A(sec²A − 1) = sec²A × tan²A (since sec²A − 1 = tan²A) = tan²A × sec²A
RHS = tan⁴A + tan²A = tan²A(tan²A + 1) = tan²A × sec²A (since 1 + tan²A = sec²A)
LHS = RHS = tan²A × sec²A ✓
Example 3: Conversion to Sine and Cosine
Prove that: cotA + tanA = secA × cosecA
Solution: LHS = cosA/sinA + sinA/cosA = (cos²A + sin²A) / (sinA × cosA) = 1 / (sinA × cosA) = cosecA × secA = RHS ✓
Example 4: Using sin²A + cos²A = 1
Prove that: √[(1 − sinA) / (1 + sinA)] = secA − tanA
Solution: LHS = √[(1 − sinA) / (1 + sinA)] × √[(1 − sinA) / (1 − sinA)] = √[(1 − sinA)² / (1 − sin²A)] = √[(1 − sinA)² / cos²A] = (1 − sinA) / cosA = 1/cosA − sinA/cosA = secA − tanA = RHS ✓
Example 5: Using cosec²A − cot²A = 1
Prove that: (cosecA + cotA)(cosecA − cotA) = 1
Solution: LHS = (cosecA + cotA)(cosecA − cotA) = cosec²A − cot²A = 1 = RHS ✓
Comparison: The Three Identities
| Identity | Trigonometric Form | Alternative Form |
|---|---|---|
| sin²A + cos²A = 1 | 1 − sin²A = cos²A | 1 − cos²A = sin²A |
| sec²A − tan²A = 1 | tan²A = sec²A − 1 | sec²A = 1 + tan²A |
| cosec²A − cot²A = 1 | cot²A = cosec²A − 1 | cosec²A = 1 + cot²A |
Common Mistakes and Fixes
| Mistake | Fix |
|---|---|
| Writing (sinA)² as sinA² | Always write sin²A for (sinA)² |
| Cancelling terms across the identity | Work on one side only — never cross the = sign |
| Forgetting to rationalise the denominator | Multiply numerator and denominator by conjugate |
| Using identities in the wrong direction | Choose the form that simplifies your expression |
| Incorrectly squaring trigonometric expressions | (sinA + cosA)² = sin²A + cos²A + 2sinAcosA = 1 + 2sinAcosA |
ICSE Exam Focus
Trigonometric identities carry 6–8 marks in ICSE exams. Questions require:
- Proving identities using the three fundamental identities.
- Expressing one ratio in terms of another.
- Simplifying complicated trigonometric expressions.
- Using algebraic techniques (factorisation, conjugates).
Marks Blueprint:
| Topic | Marks |
|---|---|
| Direct identity proof (simple) | 3 |
| Complex identity proof | 5 |
| Simplifying expressions | 2 |
| Expressing one ratio in terms of another | 2 |
Self-Test Questions
-
Prove that: √[(1 + cosA) / (1 − cosA)] = cosecA + cotA.
-
Show that: (tanA + secA − 1) / (tanA − secA + 1) = (1 + sinA) / cosA.
-
Prove that: sin⁴A − cos⁴A = sin²A − cos²A.
-
If x = a sinθ + b cosθ and y = a cosθ − b sinθ, prove that x² + y² = a² + b².
-
Prove that: (sinA − cosecA)² + (cosA − secA)² = cot²A + tan²A − 1.
-
Prove that: (1 + tan²A) / (1 + cot²A) = tan²A.
In ICSE identity proofs, work on the LHS unless the RHS is clearly more complex. Avoid moving terms across the equality sign — manipulate one side until it matches the other.
