Question

prove sinex+cosx=1

Answer 1

This is NOT a trigonometric identity to be proved. (Identities are true for all values of x.)
This is an equation that we can solve for x and it will be true only for those values of x.

Answer 2

If x = 0 then the sin and cos is zero and one respectively.
if x = 90 deg then the sin and cos is one and zero respectively.

Answer 3

In either case, the sum is one.