Question

If a cose + b sin 0 = m and a sin 8-b cos 0 = n, prove that a' + b2 = m2 +n2​

Answer 1

Answer:

Step-by-step explanation:

Given :

a cos θ + b sin θ = m,

a sin θ - b cos θ = n,

To prove :

a² + b² = m² + n²

Proof :

We know that,

sin² θ + cos² θ = 1,

______

a cos θ + b sin θ = m ...(i)

a sin θ - b cos θ = n ...(ii)

(i)² + (ii)²

⇒ (a cos θ + b sin θ)² + (a sin θ - b cos θ)² = m² + n²

⇒ a² cos² θ + b² sin² θ + 2ab sin θ cos θ + a² sin² θ + b² cos² θ - 2ab sin θ cos θ = m² + n²

⇒ a²( sin² θ + cos² θ) + b²( sin² θ + cos² θ) = m² + n²

⇒ a² (1) + b² (1) = m² + n²

⇒ a² + b² = m² + n²

Hence,proved.