Question

Given that sin θ + cos θ = x, prove that sin⁴ θ + cos⁴ θ = `(2-(x^2-1)^2)/2`.

Answer 1

(sinθ + cosθ)² = x²

(a + b)² = a² + 2ab + b²

sin²θ + 2sinθ cosθ + cos²θ = x²

Since sin⁡²θ + cos⁡²θ = 1, we substitute:

1 + 2sinθ cosθ = x²

2sinθ cosθ = x² − 1

sinθ cosθ = `(x^2-1)/2`

sin⁴ θ + cos⁴ θ

sin⁴θ + cos⁴θ = (sin²θ + cos²θ)² − 2sin²θ cos²θ

Substituting sin⁡²θ + cos²⁡θ = 1

sin⁴θ + cos⁴θ = 1 − 2sin²θ cos²θ

sin²θ cos²θ = `((x^2-1)/2)^2`

We substitute:

sin⁴θ + cos⁴θ = `1-2xx((x^2-1)^2)/4`

= `1-((x^2-1)^2)/2`

= `(2-(x^2-1)^2)/2`

sin⁴ θ + cos⁴ θ = `(2-(x^2-1)^2)/2`

Hence proved.