$$ \frac { \sin ( 180 ^ { \circ } + x ) \cdot \cos ( 180 ^ { \circ } - x ) \cdot \sin 50 ^ { \circ } } { an ( 315 ) \cdot \cos ^ { \circ } ( 360 ^ { \circ } - x ) \cdot \cos 140 ^ { \circ } } $$

Question

$$ \frac { \sin ( 180 ^ { \circ } + x ) \cdot \cos ( 180 ^ { \circ } - x ) \cdot \sin 50 ^ { \circ } } { 	an ( 315 ) \cdot \cos ^ { \circ } ( 360 ^ { \circ } - x ) \cdot \cos 140 ^ { \circ } } $$

UpStudy AI · Accepted Answer

$$
\textbf{Step 1: Simplify the numerator.}
$$

The numerator is 
$$
\sin(180^\circ+x) \cdot \cos(180^\circ-x) \cdot \sin50^\circ.
$$
Using the identities
$$
\sin(180^\circ+x)=-\sin x, \quad \cos(180^\circ-x)=-\cos x,
$$
we have
$$
\sin(180^\circ+x) \cdot \cos(180^\circ-x) = (-\sin x)(-\cos x) = \sin x \cos x.
$$
Thus, the numerator becomes
$$
\sin x \cos x \cdot \sin50^\circ.
$$

$$
\textbf{Step 2: Simplify the denominator.}
$$

The denominator is
$$
\tan315^\circ \cdot \cos(360^\circ-x) \cdot \cos140^\circ.
$$
First, evaluate $$\tan315^\circ$$. Since
$$
315^\circ = 360^\circ - 45^\circ,
$$
we have
$$
\tan315^\circ = \tan(-45^\circ) = -\tan45^\circ = -1.
$$

Next, note that
$$
\cos(360^\circ-x) = \cos x,
$$
since cosine is $$360^\circ$$-periodic.

Now, simplify $$\cos140^\circ$$ using the identity for cosine in the second quadrant:
$$
\cos140^\circ = \cos(180^\circ-40^\circ) = -\cos40^\circ.
$$

Substitute these into the denominator:
$$
\tan315^\circ \cdot \cos(360^\circ-x) \cdot \cos140^\circ = (-1) \cdot \cos x \cdot (-\cos40^\circ) = \cos x \cos40^\circ.
$$

$$
\textbf{Step 3: Combine the simplified numerator and denominator.}
$$

Now the expression becomes:
$$
\frac{\sin x \cos x \cdot \sin50^\circ}{\cos x \cos40^\circ}.
$$
Cancel the common factor $$\cos x$$ (assuming $$\cos x \neq 0$$):
$$
\frac{\sin x \sin50^\circ}{\cos40^\circ}.
$$

$$
\textbf{Step 4: Utilize a trigonometric identity.}
$$

Note that
$$
\sin50^\circ = \cos40^\circ.
$$
Substitute this into the expression:
$$
\frac{\sin x \cos40^\circ}{\cos40^\circ} = \sin x.
$$

$$
\textbf{Final Answer:}
$$

$$
\sin x.
$$
The expression simplifies to $$ \sin x $$.

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