Question

If a cos +b sin =x and a sin -b cos =y,prove that a^2+b^2=x^2+ y^2

Answer 1

Answer:

Step-by-step explanation:

x = a cosФ +b sinФ

y = a sinФ -b cosФ

R.H.S:

x² + y² = (a cosФ +b sinФ) ² + (a sinФ -b cosФ)²

= a²Cos²Ф+ b²Sin²Ф+2abCosФSinФ + a²Sin²Ф + b²Cos²Ф - 2abSinФCosФ

= a²Cos²Ф+ b²Sin²Ф + a²Sin²Ф + b²Cos²Ф

= a²(Cos²Ф + Sin²Ф) + b²(Sin²Ф + Cos²Ф)

= a² + b²

= L.H.S

Hence proved.