To prove the given trigonometric identity:
[tex]\[
\frac{1}{\sin \left(180^{\circ}+x\right) \cdot \sin \left(360^{\circ}-x\right)} - \cot^2\left(180^{\circ}-x\right) = 1
\][/tex]
Let's start using some fundamental trigonometric identities and simplify each term step-by-step.
### Step 1: Simplify [tex]\(\sin(180^\circ + x)\)[/tex]
We know the sine function has the identity:
[tex]\[
\sin(180^\circ + x) = -\sin(x)
\][/tex]
### Step 2: Simplify [tex]\(\sin(360^\circ - x)\)[/tex]
Similarly, for the sine function:
[tex]\[
\sin(360^\circ - x) = \sin(-x) = -\sin(x)
\][/tex]
### Step 3: Simplify [tex]\(\cot(180^\circ - x)\)[/tex]
Using the cotangent identity:
[tex]\[
\cot(180^\circ - x) = -\cot(x)
\][/tex]
### Step 4: Substitute the simplified terms back into the equation
Substitute [tex]\(\sin(180^\circ + x), \sin(360^\circ - x),\)[/tex] and [tex]\(\cot(180^\circ - x)\)[/tex] into the original identity:
[tex]\[
\frac{1}{\left(-\sin(x)\right) \cdot \left(-\sin(x)\right)} - \left(-\cot(x)\right)^2
\][/tex]
Since [tex]\((-1) \cdot (-1) = 1\)[/tex], we have:
[tex]\[
\frac{1}{\sin(x) \cdot \sin(x)} - (\cot(x))^2
\][/tex]
### Step 5: Simplify the left-hand side
Rewrite the left-hand side:
[tex]\[
\frac{1}{\sin^2(x)} - \cot^2(x)
\][/tex]
Recall that:
[tex]\[
\cot(x) = \frac{\cos(x)}{\sin(x)} \implies \cot^2(x) = \frac{\cos^2(x)}{\sin^2(x)}
\][/tex]
Therefore, we can express the left-hand side as:
[tex]\[
\frac{1}{\sin^2(x)} - \frac{\cos^2(x)}{\sin^2(x)}
\][/tex]
### Step 6: Combine the fractions
Combine the fractions under a common denominator:
[tex]\[
\frac{1 - \cos^2(x)}{\sin^2(x)}
\][/tex]
### Step 7: Utilize a Pythagorean identity
Recall the Pythagorean identity:
[tex]\[
\sin^2(x) + \cos^2(x) = 1 \implies 1 - \cos^2(x) = \sin^2(x)
\][/tex]
Substitute [tex]\(1 - \cos^2(x)\)[/tex] with [tex]\(\sin^2(x)\)[/tex]:
[tex]\[
\frac{\sin^2(x)}{\sin^2(x)} = 1
\][/tex]
### Step 8: Conclusion
Thus, we have shown that:
[tex]\[
\frac{1}{\sin\left(180^\circ + x\right) \cdot \sin\left(360^\circ - x\right)} - \cot^2\left(180^\circ - x\right) = 1
\][/tex]
Hence, the given identity is proven to be true.
