Simplify the following:

Step 1: Apply trigonometric reduction formulas to simplify each term in the expression. • $\sin(180^\circ - x) = \sin x$ (sine is positive in the second quadrant) • $\cos(360^\circ - x) = \cos x$ (cosine is positive in the fourth quadrant) • $\tan(180^\circ + x) = \tan x$ (tangent is positive in the third quadrant) • $\sin\frac{\pi}{2} = \sin(90^\circ) = 1$ • $\cos(180^\circ - x) = -\cos x$ (cosine is negative in the second quadrant)

Step 2: Substitute these simplified terms back into the original expression. $\frac{(\sin x)(\cos x)(\tan x)}{(1)(\cos x)(-\cos x)}$

Step 3: Simplify the numerator and the denominator. $\frac{\sin x \cos x \tan x}{-\cos^2 x}$

Step 4: Replace $\tan x$ with $\frac{\sin x}{\cos x}$. $\frac{\sin x \cos x \left(\frac{\sin x}{\cos x}\right)}{-\cos^2 x}$

Step 5: Cancel out $\cos x$ in the numerator and simplify the expression. $\frac{\sin x \sin x}{-\cos^2 x} = \frac{\sin^2 x}{-\cos^2 x}$

Step 6: Express the result in terms of $\tan x$. $-\left(\frac{\sin x}{\cos x}\right)^2 = -\tan^2 x$

The simplified expression is: $-\tan^2 x$

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