Question

if a CosØ - b SinØ = X and a SinØ + b CosØ = y. Prove that a²+b² = x²+y² ​

Answer 1

Answer:

a cosθ - b sinθ = x and a sinθ + b cosθ = y

R.H.S. = x2 + y2

= (a cosθ - b sinθ)2 + (a sinθ + b cosθ)2

= a2cos2θ - 2ab cosθ sinθ + b2sin2θ + a2sin2θ +

2absinθ cosθ + b2cos2θ

= (a2+b2) cos2θ + (b2+a2)sin2θ

= (a2+b2)cos2θ + (a2+b2)sin2θ

= (a2+b2)(cos2θ + sin2θ)

= (a2+b2) [∵ cos2θ + sin2θ = 1]

= L.H.S. ∴ a2+b2 = x2+y2.

Answer 2

Given :

a cos∅ - b sin∅ = X ...(1)

a sin∅ + b cos∅ = Y ...(2)

ON SQUARING AND ADDING BOTH THE EQUATION (1) AND (2) ..

We get -

(a cos∅ - b sin∅)² + (a sin∅ + b cos∅)² = x² + y²

a²cos²∅ + b²sin²∅ - 2abcos∅sin∅ + a²sin²∅ + b²cos²∅ + 2absin∅cos∅ = x² + y²

a²(cos²∅ + sin²∅) + b²(sin²∅ + cos²∅) = x² + y²

★ We know that { sin²∅ + cos²∅ = 1 }

Then,

a² + b² = x² + y²

hence proved