Trigonometric Identities

• Use sin²θ + cos²θ = 1 (NCERT identity).
• Derive 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ.
• Simplify expressions and prove simple identities.

1. Pythagoras identity — sin²θ + cos²θ = 1.
2. Quotient identity — tan θ = sin θ / cos θ.
3. Derived — divide first identity by cos²θ → 1 + tan²θ = sec²θ.
4. Use — express tan θ in terms of sin θ and cos θ only.
5. Proof style — start with known identity, algebraically transform both sides.

Prove sin²θ + cos²θ = 1 using a right triangle

In right ΔABC, ∠B = 90°, sin θ = AC/AB, cos θ = BC/AB.
(sin θ)² + (cos θ)² = AC²/AB² + BC²/AB²
= (AC² + BC²)/AB² = AB²/AB² = 1

• Treating sin²θ + cos²θ = 1 as sin θ + cos θ = 1 (false).
• Cancelling terms incorrectly across = sign.
• Using 1 + tan²θ = sec²θ when cos θ = 0 (θ = 90°).

• Simplify (1 − sin²θ) in terms of cos θ.
• If sin θ = 3/5, find cos θ (θ acute).
• Express tan θ in terms of sin θ only.

Open the Practice tab for graded questions on Trigonometric Identities.