can someone correct my work please? it's about trigonometry identity

[torpid raft]

Here's a softer way to do it.
Let me tell you the story of:
$$ \frac{a+b}{\frac{a}{b}-\frac{b}{a}}
= \frac{a+b}{\frac{a^2-b^2}{ab}}
= \frac{ab(a+b)}{a^2-b^2}
= \frac{ab(a+b)}{(a-b)(a+b)}= \frac{ab}{a-b}$$

Now how does this relate to your question ?
If we take $a = sin(x)$
And $b = cos(x)$ this is equal to your expression which simply becomes
$\frac{sin(x)cos(x)}{sin(x)-cos(x)}$