Consider the following: 1. sin4θ - sin2θ = cos4θ - cos2θ 2. sin4θ + cos4θ = 1 + 2sin2θ cos2θ 3. tan4θ + tan2θ = sec4θ - sec2θ  Which of the above are identities ?

Consider the following: 1. sin4θ - sin2θ = cos4θ - cos2θ 2. sin4θ + cos4θ = 1 + 2sin2θ cos2θ 3. tan4θ + tan2θ = sec4θ - sec2θ  Which of the above are identities ? Correct Answer 1 and 3 only

Formula used:

sin²x + cos²x = 1

1 + tan²x = sec²x

Calculation:

Consider 1:

sin4θ - sin2θ = cos4θ - cos2θ

⇒ sin²θ(sin²θ - 1) = cos²θ(cos²θ - 1)

⇒ sin2θ(- cos2θ) = cos2θ(- sin2θ)

Since LHS = RHS, hence true

Consider 2:

sin4θ + cos4θ = 1 + 2sin2θ cos2θ

⇒ sin4θ + cos4θ - 2sin2θ cos2θ = 1

⇒ (sin2θ - cos²θ)² = 1

Since, LHS is not equal to RHS, hence false

Consider 3:

tan4θ + tan2θ = sec4θ - sec2θ 

⇒ tan2θ(tan2θ + 1) = sec2θ(sec2θ - 1)

⇒ tan2θ sec2θ = sec2θ tan2θ

Since, LHS = RHS, hence true

∴ Only 1 and 3 are correct.