To solve the expression:
[tex]\[
\frac{\sin(180^\circ - x) \tan(90^\circ - x)}{\csc(90^\circ - x) \cos(360^\circ - x) \cot(180^\circ + x)},
\][/tex]
we will use trigonometric identities.
1. Simplify [tex]\(\sin(180^\circ - x)\)[/tex]:
Using the co-function identity:
[tex]\[
\sin(180^\circ - x) = \sin x
\][/tex]
2. Simplify [tex]\(\tan(90^\circ - x)\)[/tex]:
Using the identity:
[tex]\[
\tan(90^\circ - x) = \cot x
\][/tex]
3. Simplify [tex]\(\csc(90^\circ - x)\)[/tex]:
Using the co-function identity:
[tex]\[
\csc(90^\circ - x) = \sec x
\][/tex]
and since [tex]\(\csc x\)[/tex] is the reciprocal of [tex]\(\sin x\)[/tex], likewise [tex]\(\sec x\)[/tex] is the reciprocal of [tex]\(\cos x\)[/tex]:
[tex]\[
\csc(90^\circ - x) = \frac{1}{\cos x}
\][/tex]
4. Simplify [tex]\(\cos(360^\circ - x)\)[/tex]:
Using the co-function identity:
[tex]\[
\cos(360^\circ - x) = \cos x
\][/tex]
5. Simplify [tex]\(\cot(180^\circ + x)\)[/tex]:
Using the identity:
[tex]\[
\cot(180^\circ + x) = -\cot x
\][/tex]
Now substitute these identities back into the original expression:
[tex]\[
\frac{\sin x \cdot \cot x}{\left(\frac{1}{\cos x}\right) \cos x \cdot (-\cot x)}
\][/tex]
Simplify the numerator and denominator:
[tex]\[
= \frac{\sin x \cdot \cot x}{\left(\frac{1}{\cos x}\right) \cos x \cdot (-\cot x)}
= \frac{\sin x \cdot \cot x}{\frac{\cos x}{\cos x} \cdot (-\cot x)}
= \frac{\sin x \cdot \cot x}{1 \cdot (-\cot x)}
= \frac{\sin x \cdot \cot x}{-\cot x}
= -\sin x
\][/tex]
Thus, the final simplified expression is:
[tex]\[
-\sin x
\][/tex]
