If x = 3 sinθ + 4 cosθ and y = 3 cosθ – 4 sinθ then prove that x² + y² = 25.
OR

If sinθ + sin²θ = 1;

Thenprovethatcos²θ+cos⁴θ= 1.

Question

If x = 3 sinθ + 4 cosθ and y = 3 cosθ – 4 sinθ then prove that x2 + y2 = 25.ORIf sinθ + sin2θ = 1;Thenprovethatcos2θ+cos4θ= 1.

Philoid · Accepted Answer

Given, x = 3 sin θ + 4 cos θ y = 3 cos θ – 4 sin θ

Taking LHS

x² + y² = (3 sin θ + 4 cos θ)² + (3 cos θ – 4 sin θ)²

= 9sin²θ + 16cos²θ + 24sinθcosθ + 9cos²θ

+ 16sin²θ – 24sinθcosθ

= 9(sin²θ + cos²θ) + 16(sin²θ + cos²θ)

= 9 + 16 = 25 = RHS

OR

Given, sin θ + sin²θ = 1

⇒ sin θ = 1 – sin²θ

⇒ sin θ = cos²θ [1]

Taking LHS

cos²θ + cos⁴θ

= cos²θ + sin²θ [From 1]

= 1 = RHS