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Simplify the following:

August 1, 2026Mathematics

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Answer

-\tan^2 x

6 steps

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)
Substitute these simplified terms back into the original expression.
$\frac{(\sin x)(\cos x)(\tan x)}{(1)(\cos x)(-\cos x)}$
Simplify the numerator and the denominator.
$\frac{\sin x \cos x \tan x}{-\cos^2 x}$
Replace $\tan x$ with $\frac{\sin x}{\cos x}$ .
$\frac{\sin x \cos x \left(\frac{\sin x}{\cos x}\right)}{-\cos^2 x}$
Cancel out $\cos x$ in the numerator and simplify the expression.
$\frac{\sin x \sin x}{-\cos^2 x} = \frac{\sin^2 x}{-\cos^2 x}$
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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