The general solution of the equation $${\sin ^{50}}x - {\cos ^{50}}x = 1{\text{ is}}$$

Solution :
We have,
$$\eqalign{ & {\sin ^{50}}x - {\cos ^{50}}x = 1 \cr & \Rightarrow {\sin ^{50}}x = 1 + {\cos ^{50}}x \cr} $$
Since, $${\sin ^{50}}x \leqslant 1$$   and $$1 + {\cos ^{50}}x \geqslant 1$$    therefore, the two sides are equal only if
$$\eqalign{ & {\sin ^{50}}x = 1 = 1 + {\cos ^{50}}x{\text{ i}}{\text{.e }}{\sin ^{50}}x = 1\,{\text{and }}{\cos ^{50}}x = 0 \cr & \therefore x = 2n\pi + \frac{\pi }{2},n \in I. \cr} $$