Use complex sine and cosine. 方程 cos x + sin x = 2 有无穷个解。

Use

Sin x = (e^{ix} - e^{- ix})/2i 

Cos x =  (e^{ix} + e^{- ix})/2

to transform 

cos x + sin x = 2

into a quadratic equation in e^{ix}.

Solve the quadratic equation for e^{ix} and then take complex log.

It will give infinitely many complex roots.