1. Simplify the expression $(\sin 2 x-2 \cos x) \frac{\tan x}{1-\sin ^{2} x} \cdot(1+\sin x)$.
1. We use the identity for $\sin 2 x=2 \sin x \cdot \cos x$ and then factor out the common factor $2 \sin x(\sin x+1)$ from the first factor. We simplify the second factor $\frac{\tan x}{1-\sin ^{2} x}=\frac{\frac{\sin x}{\cos x}}{(1-\sin x)(1+\sin x)}$. We simplify the expression and get $-2 \sin x$.
Expression for the sine of a double angle $\sin 2 x=2 \sin x \cdot \cos x$ ..... 1 point
Factoring out the common factor $2 \sin x(\sin x+1)$ ..... 1 point
Simplifying the second factor $\frac{\tan x}{1-\sin ^{2} x}=\frac{\frac{\sin x}{\cos x}}{(1-\sin x)(1+\sin x)}$ ..... 1 point
Simplification ..... 2 points
Result $-2 \sin x$ ..... 1 point