Expert: Sherry Wallin - 7/14/2011

Question
If x=asinx+bcosx,
  y=acosx-bsinx,
Then show that
(ax+by)^2+(bx+ay)^2=(a^2+b^2)^2

Answer
Dhananjay~

I think you must have a typo because I do not believe this is true the way you have it. Pretend that the angle is pi/6 then you would have:
x=(1/2)a + (sqrt3/2)b  and y = (sqrt3/2)a -(1/2)b

Now (ax+by)^2 + (bx+ay)^2 = (a^2+b^2) + 4abxy =>
(a^2+b^2) + 4ab((1/2)a +(sqrt3/2))(sqrt3/2)a -(1/2)b) =>
(a^2+b^2) + a^3bsqrt3 +2a^2b^2-ab^3sqrt3 which does not equal a^4+2a^b^2+b^4

Keep in mind for a statement to be true it must be true for angle values. The other thing that disturbs me is that you are using x in two different contexts and this should never be done. I suspect you used x for theta like we have done in the past but do no do that when you've named x already, use some other name like z perhaps...

If you have found you wrote the problem wrong let me know and I will be happy to help you but as it sits it just is not true. And to show something is not the case all you have to do is find one case where it doesn't work and that is what I have done for you above.

Math Prof