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