Question

if a cos theta + b sin theta equal to M and a sin theta minus B cos theta equal to N prove that n square + n square equal to a square + b square​

Answer 1

Given: a cos θ + b sin θ = m …….(1)

a sin θ – b cos θ = n …….(2)

Square equation (1) and (2) on both sides:

a²cos²θ + b²sin²θ + 2ab cos θ sin θ = m² …….(3)

a²sin²θ + b²cos²θ – 2ab cos θ sin θ = n² ……..(4)

Add equation (3) and (4):

[a²cos²θ + b²sin²θ + 2ab cos θ sin θ] + [a²sin²θ + b²cos²θ – 2ab cos θ sin θ] = m² + n²

⇒ a²(cos²θ + sin²θ) + b²(cos²θ + sin²θ) = m² + n²

⇒ a² + b² = m² + n²

Hence, proved.

Answer 2

Answer:

Step-by-step explanation:

Given: a cos θ + b sin θ = m …….(1)

a sin θ – b cos θ = n …….(2)

Square equation (1) and (2) on both sides:

a²cos²θ + b²sin²θ + 2ab cos θ sin θ = m² …….(3)

a²sin²θ + b²cos²θ – 2ab cos θ sin θ = n² ……..(4)

Add equation (3) and (4):

[a²cos²θ + b²sin²θ + 2ab cos θ sin θ] + [a²sin²θ + b²cos²θ – 2ab cos θ sin θ] = m² + n²

⇒ a²(cos²θ + sin²θ) + b²(cos²θ + sin²θ) = m² + n²

⇒ a² + b² = m² + n²

Hence, proved.

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