Question

If x sin³ θ + y cos³ θ = sin θ cos θ and x sin θ = y cos θ, then prove that x² + y² = 1

Answer 1

Given x sin² θ + y cos² θ = sin θ cos θ

x sin θ = y cos θ  ...(1)

x sin³ θ + y cos³ θ = sin θ cos θ

x sin θ (sin² θ) + y cos θ (cos² θ) = sin θ cos θ

x sin θ (sin² θ) + x sin θ (cos² θ) = sin θ cos θ

x sin θ (sin² θ + cos² θ) = sin θ cos θ

x sin θ = sin θ cos θ

x = cos θ

substitute x = cos θ in (1)

cos θ sin θ = y cos θ y = sin θ

L.H.S = x² + y² = cos² θ + sin² θ = 1

L.H.S = R.H.S

Hence it is proved.