ICSE Questions & Answers: Maths - Trigonometric Identities

• Class 10 Maths

1. Prove the standard identity sec²θ - tan²θ = 1 by using sin²θ + cos²θ = 1?

LHS = sec²θ - tan²θ

= 1/cos²θ - sin²θ/cos²θ

= (1 - sin²θ)/cos²θ

= cos²θ/cos²θ

= 1

2. Prove that: sin⁴θ + cos⁴θ = 1 - 2sin²θcos²θ

We know sin²θ + cos²θ = 1

Squaring both sides

(sin²θ + cos²θ)² = 1²

sin⁴θ + 2sin²θcos²θ + cos⁴θ = 1

sin⁴θ + cos⁴θ = 1 - 2sin²θcos²θ

3. Prove: cos²θ(1 + tan²θ) = 1

LHS = cos²θ(1 + tan²θ)

= cos²θ(sec²θ)

= cos²θ (1/cos²θ)

= 1 = RHS

4. Prove: (1 - tan A)² + (1 + tan A)² = 2 sec²A

LHS = (1 - tan A)² + (1 + tan A)²

= (1 - 2tan A + tan²A) + (1 + 2tan A + tan²A)

= 2 + 2tan²A

= 2(1 + tan²A)

= 2 sec²A

= RHS

5. Prove: tan²x(1 + cot²x) = 1/(1 - sin²x)

LHS = tan²x(1 + cot²x)

= tan²x + tan²x cot²x

= tan²x + tan²x (1/tan²x)

= tan²x + 1

= sec²x

= 1/cos²x

= 1/(1 - sin²x) = RHS