Question

$\sf{ {sin}^{4} \: \theta \: - {cos}^{4} \: \theta = 1 - 2 \: {cos}^{2} \: \theta}$

Prove it ​​

Answer 1

Required Answer:-

Given to prove:

• sin⁴ θ - cos⁴ θ = 1 - 2cos² θ

Proof:

Taking LHS,

We know that,

➡ a² - b² = (a + b)(a - b)

Therefore,

sin⁴ θ - cos⁴ θ

= (sin² θ + cos² θ)(sin² θ - cos² θ)

We also know that,

➡ sin² θ + cos² θ = 1

Therefore,

= 1 × (sin² θ - cos² θ)

= sin² θ - cos² θ

As,

➡ sin² θ + cos² θ = 1

➡ sin² θ = 1 - cos² θ

Substituting the value, we get,

= 1 - cos² θ - cos² θ

= 1 - 2cos² θ

= RHS (Hence Proved)

Formulae Used:

• sin² θ + cos² θ = 1

Answer 2

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