Answer :
The expression simplifies to a final value of 1. This is achieved through trigonometric identities and simplification of each term in the given expression.
Here is the step-by-step breakdown of the given expression:
[tex]\[ \frac{\cos(90\°+x) \cdot \sin(180\°+x)}{\tan 225\° - \cos^2(-x)} \][/tex]
Simplify each trigonometric term:
cos(90°+x):
Using the trigonometric identity, cos(90°+x) = -sin(x).
So, cos(90°+x) = -sin(x).
sin(180°+x):
Using the trigonometric identity, sin(180°+x) = -sin(x).
So, sin(180°+x) = -sin(x).
tan 225°:
225° is in the third quadrant where tangent is positive. Using the trigonometric identity, tan(225°) = tan(180°+45°) = tan(45°) = 1.
So, tan(225°) = 1.
cos²(-x):
Cosine is an even function, so cos(-x) = cos(x).
Therefore,cos²(-x) = cos²(x).
Substitute the simplified values into the expression:
[tex]\[ \frac{(-\sin(x)) \cdot (-\sin(x))}{1 - \cos^2(x)} \][/tex]
Simplify the numerator:
-sin(x) . -sin(x)) = sin²(x)
Rewrite the denominator:
Using the Pythagorean identity, 1 - cos²(x) = sin²(x).
So, the expression becomes:
[tex]\[ \frac{\sin^{2} (x)}{\sin^{2} (x)} \][/tex]
Simplify the fraction:
Since [tex]\(\frac{\sin^{2} (x)}{\sin^{2} (x)} = 1\).[/tex]
Therefore, the final simplified value of the given expression is:
[tex]\(\boxed{1}\)[/tex]
