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To solve this problem, let's go through each part systematically, using the given information and trigonometric identities. 

Given:
\[
\tan \theta = 5
\]

We need to find exact values for:
1. \( \sec^2 \theta \)
2. \( \cot \theta \)
3. \( \cot \left( \frac{\pi}{2} - \theta \right) \)
4. \( \csc^2 \theta \)

### Step 1: Find \( \sec^2 \theta \)
Using the identity:
\[
\sec^2 \theta = 1 + \tan^2 \theta
\]
we can substitute \( \tan \theta = 5 \):

\[
\sec^2 \theta = 1 + 5^2 = 1 + 25 = 26
\]

So, \( \sec^2 \theta = 26 \), which matches the answer given in part (a).

### Step 2: Find \( \cot \theta \)
The cotangent function is the reciprocal of tangent:
\[
\cot \theta = \frac{1}{\tan \theta}
\]
Substituting \( \tan \theta = 5 \):

\[
\cot \theta = \frac{1}{5}
\]

So, \( \cot \theta = \frac{1}{5} \).

### Step 3: Find \( \cot \left( \frac{\pi}{2} - \theta \right) \)
Using the co-function identity:
\[
\cot \left( \frac{\pi}{2} - \theta \right) = \tan \theta
\]
Since \( \tan \theta = 5 \), we have:

\[
\cot \left( \frac{\pi}{2} - \theta \right) = 5
\]

### Step 4: Find \( \csc^2 \theta \)
Using the identity:
\[
\csc^2 \theta = 1 + \cot^2 \theta
\]
Substituting \( \cot \theta = \frac{1}{5} \):

\[
\csc^2 \theta = 1 + \left( \frac{1}{5} \right)^2 = 1 + \frac{1}{25} = \frac{26}{25}
\]

### Summary of Answers
- (a) \( \sec^2 \theta = 26 \)
- (b) \( \cot \theta = \frac{1}{5} \)
- (c) \( \cot \left( \frac{\pi}{2} - \theta \right) = 5 \)
- (d) \( \csc^2 \theta = \frac{26}{25} \)

Would you like further explanation on any of these steps?

Here are some related questions that might deepen your understanding:
1. How do trigonometric identities help in simplifying expressions?
2. Why is the identity \( \sec^2 \theta = 1 + \tan^2 \theta \) useful in trigonometry?
3. What is the co-function identity for other trigonometric functions?
4. How does the reciprocal identity relate \(\tan\) and \(\cot\)?
5. Can we use these identities to solve real-world problems involving angles?

**Tip:** Always double-check the identities and values you substitute to ensure accuracy in trigonometric problems.

Given tan(θ) = 5, use trigonometric identities to find the exact values of sec^2(θ), cot(θ), cot(π/2 - θ), and csc^2(θ).

Find exact values of trigonometric functions sec^2(θ), cot(θ), cot(π/2 - θ), and csc^2(θ) using identities and given tan(θ) = 5. This problem explores trigonometric identities and is suitable for students in Grades 10-12.

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