if cos theta + sin theta =root2 cos theta prove that cos theta - sin theta =root 2 sin theta
let theta = x
cos x + sin x = root2 cos x
squaring on both side, we get......
cos2x + sin2x + 2cosxsinx = 2cos2x
2sinxcosx = 2cos2x - cos2x - sin2x
2sinxcosx = cos2x - sin2x
2sinxcosx = (cosx+sinx) (cosx - sinx)
2sinxcosx = (root2 cosx) (cosx - sinx)
2sinxcosx/root2 cosx = cosx - sinx
root2 sinx = cosx - sinx