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The point is that 2 is the maximal value of the given function.
It can be proved by standard method of Calculus:
by taking the derivative, equating it to zero, and then solving the associated equation.
In this post I will show you another way, which quickly leads to the answer.
(1) First notice that cos(x)+sin(x) has the maximum value of at x = . // Every Calculus student must know it (!)
(2) At this value of x, sin(2x) = 1 and, therefore, sin(2x)+1 = 2. It is maximum value of sin(2x)+1.
(3) Therefore, at x= = = 2.
(4) At all other values of x in the interval [,), cos(x) + sin(x) is less than
and sin(2x)+1 is less than 2; therefore, the entire function is strictly less than 2.
(5) Thus, the only solution to the original equation in the interval [,) is x= .
Solved.