What is the principal solution of sinx=siny

Question

X=nπ+(-1)^ny prove that 

Answer 1

Proof: Consider the equation, sin x = sin y. Let us try to find the general solution for this trigonometric equation.

sin x = sin y

⇒ sin x – sin y = 0

⇒2cos (x + y)/2 sin (x – y)/2 = 0

⇒cos (x + y)/2 = 0 or sin (x – y)/2 = 0

Upon taking the common solution from both the conditions, we get:

x = nπ + (-1)ⁿy, where n ∈ Z