1+sin x/cos x - cos x/sin x - 1

The trick is to make the denominators the same as if you were adding any ol' fraction.

\frac{(sin(x)-1)}{(sin(x)-1)}\cdot\frac{(1+sin(x))}{cos(x)}-\frac{cos(x)}{(sin(x)-1)}\cdot\frac{cos(x)}{cos(x)}

Do this and expanding, we get \frac{sin^{2}(x)-1-cos^{2}(x)}{cos(x)(sin(x)-1)}

But sin^{2}(x)-1=-cos^{2}(x)

Can you continue and whittle away at it some more?