To simplify the expression [tex]\(\tan \left(\frac{9\pi}{2} - x\right) \cdot \cot(x)\)[/tex], we'll go through a series of trigonometric identities and transformations. Here's the step-by-step solution:
1. Recall Trigonometric Identities:
- [tex]\(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)[/tex]
- [tex]\(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\)[/tex]
2. Write the Given Expression Using these Identities:
[tex]\[
\tan \left(\frac{9\pi}{2} - x\right) \cdot \cot(x)
\][/tex]
Substitute [tex]\(\tan\)[/tex] and [tex]\(\cot\)[/tex]:
[tex]\[
= \left(\frac{\sin\left(\frac{9\pi}{2} - x\right)}{\cos\left(\frac{9\pi}{2} - x\right)}\right) \cdot \left(\frac{\cos(x)}{\sin(x)}\right)
\][/tex]
3. Simplify the Expression:
Combine the fractions:
[tex]\[
= \frac{\sin\left(\frac{9\pi}{2} - x\right) \cdot \cos(x)}{\cos\left(\frac{9\pi}{2} - x\right) \cdot \sin(x)}
\][/tex]
4. Simplify the Argument:
Recall that the trigonometric functions have periods:
- [tex]\(\sin\)[/tex] has a period of [tex]\(2\pi\)[/tex]
- [tex]\(\cos\)[/tex] has a period of [tex]\(2\pi\)[/tex]
Notice that:
[tex]\[
\frac{9\pi}{2} = 4\pi + \frac{\pi}{2} = 2 \cdot 2\pi + \frac{\pi}{2}
\][/tex]
Therefore,
[tex]\[
\sin\left(\frac{9\pi}{2} - x\right) = \sin\left(\pi/2 - x\right)
\][/tex]
[tex]\[
\cos\left(\frac{9\pi}{2} - x\right) = \cos\left(\pi/2 - x\right)
\][/tex]
5. Apply Trigonometric Identities:
Recall that:
[tex]\[
\sin\left(\pi/2 - x\right) = \cos(x)
\][/tex]
[tex]\[
\cos\left(\pi/2 - x\right) = \sin(x)
\][/tex]
Substitute these back into the expression:
[tex]\[
\frac{\sin\left(\pi/2 - x\right) \cdot \cos(x)}{\cos\left(\pi/2 - x\right) \cdot \sin(x)} = \frac{\cos(x) \cdot \cos(x)}{\sin(x) \cdot \sin(x)}
\][/tex]
6. Simplify the Final Expression:
[tex]\[
= \frac{\cos^2(x)}{\sin^2(x)}
\][/tex]
7. Final Answer:
Using the identity [tex]\(\cot(x) = \frac{\cos(x)}{\sin(x)}\)[/tex], we can write:
[tex]\[
= \cot^2(x)
\][/tex]
Thus, the simplified form of [tex]\(\tan \left(\frac{9\pi}{2} - x\right) \cdot \cot(x)\)[/tex] is:
[tex]\[
\boxed{\cot^2(x)}
\][/tex]
