Simplify the following expression as much as possible:

[tex]\[ \tan \left(\frac{9 \pi}{2} - x\right) \][/tex]

[tex]\[ \cot (x) \][/tex]



Answer :

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]

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