# $Explaintheproofoftheorem:Foranyrealnumbersxandy,\mathrm{sin}x=\mathrm{sin}yimpliesx=n\mathrm{\pi}+{\left(-1\right)}^{\mathrm{n}}\mathrm{y},\mathrm{where}\mathrm{n}\in \mathrm{Z}$

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!!

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