Math can be so interesting sometimes.

Let's say you want to simplify this:

$$ \frac{\sin^2 x}{1 - \cos x} - 1 $$

One path

I thought, aha!, there's a 1 there! And I know that $\sin^2x + \cos^2 = 1 $. So I could just do this right?

$$ \frac{\sin^2 x}{1 - \cos x} - \sin^2x + \cos^2 $$

And from here you could expand and try something like:

$$ \frac{\sin^2 x}{1 - \cos x} - \frac{(1 - \cos x)(\sin^2x + \cos^2)}{1 - \cos x} $$

You now have something with the same denominator and this would get you there eventually ... but only if you're completely sure that you don't make any manual mistakes.

Another path

Instead you could also rewrite $\sin^2x + \cos^2 = 1 $ into $\cos^2 x = 1 - \sin^2x $. Why would this help? Well let's go back to the start.

$$ \frac{\sin^2 x}{1 - \cos x} - 1 $$

This becomes.

$$ = \frac{1 - \cos^2 x}{1 - \cos x} - 1 $$

You can factor this.

$$ = \frac{(1 - \cos x)(1 + \cos x)}{1 - \cos x} - 1 $$

And hey! Notice how things cancel out pretty early here!

$$ = \frac{(\cancel{1 - \cos x})(1 + \cos x)}{\cancel{1 - \cos x}} - 1 $$

That makes things a lot simpler.

$$ = 1 + \cos x - 1 $$

$$ = \cos x $$

The tricky part with treating math as an occasional hobby, instead of a daily activity, is that you get very rusty at detecting the right path early.