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