$Explain the proof of theorem: For any real numbers x and y, \sin x=\sin y implies x=n\mathrm{\pi }+{\left(-1\right)}^{\mathrm{n}}\mathrm{y}, \mathrm{where} \mathrm{n}\in \mathrm{Z}$

From the above image, we have

Now, y has coefficient -1, which can be written as (-1)²ⁿ⁺¹ for n ∈ Z.

As for n = 1, we have (-1)²ⁿ⁺¹ = (-1)^2.1+1 = (-1)³ = -1

similarly, for n = 2, 3 and so on (-1)²ⁿ⁺¹ = -1 because (2n + 1) is an odd number. 

So, we write (-1)²ⁿ⁺¹ on the place of -1 as:

Again, from the image shown above, it is clear that

Again, y has a coefficient equal to +1.

Now, +1 can be written as: (-1)²ⁿ where n ∈ Z.

As for n = 1, we have (-1)²ⁿ = (-1)^2.(1) = (-1)² = 1

similarly, for n = 2, 3 and so on (-1)²ⁿ = 1 because 2n is an even number. 

So, we get

The joining of two equation is illustrated as:

Hope you get it!!