The NCERT Math Textbook for Grade 11 mentions these two general solutions for Cosine Trigonometric function:
$\cos x = 0$ gives $x=(2n+1)\pi/2$, where $n\in Z$
$\cos x = \cos y$ gives $x=2n\pi \pm y$, where $n\in Z$
So if I have to solve
$\cos x = 0$
why can't I simplify it as
$\cos x = \cos \pi/2$
and use the second formula to say
$x=2n\pi \pm \pi/2$
I understand that both the solutions cover all the odd multiples of $\pi/2$ for different values of $n$, but I've not come across even one example that solves $\cos x = 0$ using the second solution
What do you mean? Just factor out $π/2,$ to get $$x=π/2(4n\pm 1).$$ The last factor contains all odd numbers since $4n+3$ can be taken instead of $4n-1$ (this is just a shift), and all odd numbers are of either of the forms, since $4n,4n+2$ can never be odd.
So the solutions are actually equivalent, only apparently different.