Question

If a cos θ + b sin θ = m and a sin θ – b cos θ = n, prove that a² + b² = m² + n²

Answer 1

R.H.S `m^2 sin^2 theta`

`= (a cos theta + b sin theta)^2 + (a sin theta - b cos theta)^2`

`= a^2 cos^2 theta + b^2 sin^2 theta + 2 ab sin theta cos theta + a^2 sin^2 theta + b^2 cos^2 theta - 2 ab sin theta cos theta`

`= a^2 cos^2 theta + b^2 cos^2 theta + b^2 sin^2 theta + a^2 sin^2 theta`

`= a^2(sin^2 theta + cos^2 theta) + b^2(sin^2 theta + cos^2 theta)`

`=a^2 + b^2`   (∵ `sin^2 theta + cos^2 theta = 1`)