Solve the following equation and state the general solution for all values of x in exact form. Show all steps of your algebraic solution.

this is about the 3rd time this has come up, so again,

since sin^2x = 1-cos^2x
you have
1 - cos^2x - √2 cosx = cos^2x + √2 cosx + 2
2cos^2x + 2√2 cosx + 1 = 0
(√2 cosx + 1)^2 = 0
cosx = -1/√2
x = π ± π/4 + 2kπ