Question

if cos^-1x+cos^-1y+cos^-1z=π,prove that x^2+y^2+z^2+2xyz=1​

Answer 1

Answer:

Step-by-step explanation: We use a classical infinite descent argument.

Note that the right-hand side is even, so the left-hand side must be. It follows that two of x, y, z are odd and the third even, or all three are even.

But two odd and one even is impossible, for then the right-hand side is divisible by 4 and the left-hand side is not.

Thus x=2x1, y=2y1, z=2z1 for some integers x1, y1, z1.

Substituting we get x21+y21+z21=4x1y1z1.

Repeat the argument. We find that x1=2x2, and so on, with x22+y22+z22=8x2y2z2.

Continue. We conclude that x, y, z are each divisible by arbitrarily high powers of 2, so are all 0.