The given expression can be simplified step by step by applying trigonometric identities. Let's break it down and simplify:

Expression:

$\frac{\sin\left(\frac{\pi}{2} - x\right) + \cos(-x) + \sin(2\pi - x) + \cos\left(\frac{\pi}{2} - x\right)}{\cos\left(\frac{\pi}{2} + x\right) + \sin(-x)}$

Step 1: Apply trigonometric identities

• $\sin\left(\frac{\pi}{2} - x\right) = \cos(x)$ (co-function identity)
• \cos(-x) = \cos(x) (cosine is even)
• \sin(2\pi - x) = -\sin(x)) (periodic identity of sine)
• \cos\left(\frac{\pi}{2} - x\right) = \sin(x)) (co-function identity)
• \cos\left(\frac{\pi}{2} + x\right) = -\sin(x)) (co-function identity)
• \sin(-x) = -\sin(x) (sine is odd)

Step 2: Substitute these identities into the expression

Numerator:

$\sin\left(\frac{\pi}{2} - x\right) + \cos(-x) + \sin(2\pi - x) + \cos\left(\frac{\pi}{2} - x\right)$

becomes

$\cos(x) + \cos(x) - \sin(x) + \sin(x)$

Simplifying this:

$2\cos(x)$

Denominator:

$\cos\left(\frac{\pi}{2} + x\right) + \sin(-x)$

becomes

$-\sin(x) - \sin(x)$

Simplifying this:

$-2\sin(x)$

Step 3: Final expression

$\frac{2\cos(x)}{-2\sin(x)} = -\cot(x)$

Final Answer:

$-\cot(x)$

Would you like more details or have any questions?

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Here are 5 follow-up questions you might explore:

1. What is the