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I am working on trigonometric theorems from this book. I am referring to Theorem $1$ on page $75$. I am not able to understand how final conclusion is derived from second last step. I am not able to derive the highlighted part from the $2$ equations above it.

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    Every integer must be either odd or even ... – 2017-01-26
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    I see that now. Thanks. – 2017-01-26
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    Please post as an answer so that I can mark it as solution. – 2017-01-26

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If you combine $x=2n\pi$ with $x=(2n+1)\pi$, you get $x=n\pi$ for all $n\in\textbf{Z}=\{2n+1\}\cup \{2n\}$

Likewise $(-1)^{2n+1}$ and $(-1)^{2n}$ combine to give $(-1)^n$ for all $n\in\textbf{Z}$ which gives the result. Combination here means union.

The solution $x=n\pi+(-1)^n y$, $n\in\textbf{Z}$, gives all the possible values from $x=2n\pi+(-1)^{2n}$ and $x=(2n+1)\pi+(-1)^{2n+1}$