Question

If a cosx+b sinx = p and a sinx - b cosx =q, show that a^2 + b^2 = p^2 + q^2

Answer 1

acosx+bsinx=p
or, p²=a²cos²x+2absinxcosx+b²sin²x and
asinx-bcosx=q
or, q²=a²sin²x-2absinxcosx+b²cos²x
∴, p²+q²=a²(sin²x+cos²x)+2absinxcosx-2absinxcosx+b²((sin²x+cos²x)
or, p²+q²=a²+b² [∵, sin²x+²cos²x=1]