REDUZCA:
 sen⁡(π/2+x)/cos⁡(π-x) +tan⁡(π-x┤/cos⁡(3π/2-x) 


Let's simplify the expression:

\[
E = \frac{\sin\left(\frac{\pi}{2} + x\right)}{\cos(\pi - x)} + \frac{\tan(\pi - x)}{\cos\left(\frac{3\pi}{2} - x\right)}
\]

### Step 1: Simplify \( \sin\left(\frac{\pi}{2} + x\right) \)
Using the identity \( \sin\left(\frac{\pi}{2} + x\right) = \cos(x) \), we have:

\[
\sin\left(\frac{\pi}{2} + x\right) = \cos(x)
\]

### Step 2: Simplify \( \cos(\pi - x) \)
Using the identity \( \cos(\pi - x) = -\cos(x) \), we get:

\[
\cos(\pi - x) = -\cos(x)
\]

### Step 3: Simplify the first term
Substituting the simplified terms into the first part of the expression:

\[
\frac{\sin\left(\frac{\pi}{2} + x\right)}{\cos(\pi - x)} = \frac{\cos(x)}{-\cos(x)} = -1
\]

### Step 4: Simplify \( \tan(\pi - x) \)
Using the identity \( \tan(\pi - x) = -\tan(x) \), we have:

\[
\tan(\pi - x) = -\tan(x)
\]

### Step 5: Simplify \( \cos\left(\frac{3\pi}{2} - x\right) \)
Using the identity \( \cos\left(\frac{3\pi}{2} - x\right) = -\sin(x) \), we get:

\[
\cos\left(\frac{3\pi}{2} - x\right) = -\sin(x)
\]

### Step 6: Simplify the second term
Substituting the simplified terms into the second part of the expression:

\[
\frac{\tan(\pi - x)}{\cos\left(\frac{3\pi}{2} - x\right)} = \frac{-\tan(x)}{-\sin(x)} = \frac{\tan(x)}{\sin(x)}
\]

Since \( \frac{\tan(x)}{\sin(x)} = \frac{\sin(x)/\cos(x)}{\sin(x)} = \frac{1}{\cos(x)} = \sec(x) \), the second term simplifies to \( \sec(x) \).

### Final Simplified Expression
Combining everything:

\[
E = -1 + \sec(x)
\]

So, the reduced expression is:

\[
E = \sec(x) - 1
\]

Would you like more details or have any questions?

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Here are 5 related questions you might find interesting:
1. What are the other co-function identities and how can they be applied?
2. How do the Pythagorean identities relate to trigonometric functions?
3. What is the geometric interpretation of the secant function?
4. How do you derive the identities used in trigonometric simplifications?
5. What are the general methods for simplifying complex trigonometric expressions?

**Tip:** Familiarity with trigonometric identities can significantly simplify complex expressions. Consider practicing these identities regularly.

Learn how to simplify the trigonometric expression sin((π/2) + x)/cos(π - x) + tan(π - x)/cos((3π/2) - x) step-by-step. This problem involves applying trigonometric identities such as co-function identities and Pythagorean identities. Suitable for advanced high school students and anyone studying trigonometry.

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