Here is the solution for the given expressions written in terms of simple trigonometric functions:

a. $\text{cosec}(-x)$

Using the property $\text{cosec}(-x) = -\text{cosec}(x)$, we have: $\text{cosec}(-x) = -\text{cosec}(x)$

b. $\text{cot}(-x)$

Using the property $\text{cot}(-x) = -\text{cot}(x)$, we have: $\text{cot}(-x) = -\text{cot}(x)$

c. $\cos(-x)$

Using the property $\cos(-x) = \cos(x)$, we have: $\cos(-x) = \cos(x)$

d. $\sin\left(\frac{\pi}{2} + x\right)$

Using the property $\sin\left(\frac{\pi}{2} + x\right) = \cos(x)$, we have: $\sin\left(\frac{\pi}{2} + x\right) = \cos(x)$

e. $\cos(\pi - x)$

Using the property $\cos(\pi - x) = -\cos(x)$, we have: $\cos(\pi - x) = -\cos(x)$

f. $\tan\left(\frac{\pi}{2} - x\right)$

Using the property $\tan\left(\frac{\pi}{2} - x\right) = \cot(x)$, we have: $\tan\left(\frac{\pi}{2} - x\right) = \cot(x)$

Let me know if you want further clarification or have any specific questions about these identities!

Follow-up questions for better understanding:

1. Why does the sine function change to cosine in the expression $\sin\left(\frac{\pi}{2} + x\right)$?
2. How do the even/odd properties of trigonometric functions affect these transformations?
3. Can you derive the identity for $\tan\left(\frac{\pi}{2} - x\right)$ using the sine and cosine ratio?
4. Why does $\cos(-x)$ remain unchanged, while $\text{cot}(-x)$ changes sign?
5. How can these properties be used in solving equations involving multiple angles?

Tip: Always remember the quadrant system when dealing with trigonometric transformations; it helps in determining signs and relationships of the functions.