Question

\\\ln x \cdot x^{\cos(x)} \cdot -\sin x\\
\\-\sin x \ln x \cdot x^{\cos(x)}\\

Explanation:

Identify the mathematical expression

We analyze the handwritten expression in the image.
The first line reads:

$$ \ln x \cdot x^{\cos(x)} \cdot -\sin x $$

The second line reads:

$$ -\sin x \ln x \cdot x^{\cos(x)} $$

These represent two equivalent algebraic expressions.

Simplify the expression

We rearrange the factors to write the expression in standard mathematical notation.
By moving the negative sign and the trigonometric function to the front:

$$ -\sin(x) \ln(x) x^{\cos(x)} $$

Verify equivalence of both lines

We check if the second line is indeed the simplified version of the first line.
First line:

$$ (\ln x \cdot x^{\cos(x)}) \cdot (-\sin x) $$

Using the commutative property of multiplication:

$$ = -\sin x \cdot \ln x \cdot x^{\cos(x)} $$

This matches the second line exactly.

Answer:

$$ -\sin(x) \ln(x) x^{\cos(x)} $$