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While going through 'Plane Trigonometry by SL Loney' I came across an article for general value of $\cos x$ which is $\left(2n\pm\frac12\right)\cdot\frac\pi2$. But, when I am solving $\cos x = 0$, the answer I am getting is $\left(2n\pm\frac12\right)\cdot\frac\pi2$, but as per book the answer is $\left(n+\frac12\right)\cdot\frac\pi2$. Why does the answer only contain the plus sign but not the minus sign?

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Both of your answers are correct, and are different representations of each other.

We can easily show that $\frac12\left(2m\pm\frac12\right)=\frac12\left(n+\frac12\right)$ for integer $m,n$. For any $m$, if $n=2m$, $\frac12\left(2m+\frac12\right)=\frac12\left(n+\frac12\right)$ (note the + sign) and if $n=2m-1$ then $\frac12\left(2m-\frac12\right)=\frac12\left(n+\frac12\right)$.

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If $\dfrac{(2n_1+1)\pi}2=\dfrac{(2n_2-1)\pi}2$

$\iff2n_1+1=2n_2-1\iff n_1=n_2-1$

So, both $\dfrac{(2n_1+1)\pi}2,\dfrac{(2n_2-1)\pi}2$ independently covers the exhaustive solution set for integer values of $n_1,n_2$

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