Explanation

Note that 2tan x/(1 + tan²x) = sin 2x, so the equation becomes (1 − tan x)/(1 + tan x) = 1 − sin 2x. Writing the left side as tan(π/4 − x) and substituting t = π/4 − x gives tan t = 1 − cos 2t = 2sin²t, i.e. sin t(1/cos t − 2 sin t) = 0. Since x ≠ π/4, sin 2t = 1 gives t = π/4 and x = 0. Then sin x + cos x = 1, option C.