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The problem asks to find \( \tan^2(\theta) + \cot^2(\theta) \) given that \( \tan(\theta) + \cot(\theta) = 5 \).

Let's solve it step by step.

### Step 1: Define terms
We know:
\[
\tan(\theta) + \cot(\theta) = 5
\]
Also, we can express \( \cot(\theta) \) as:
\[
\cot(\theta) = \frac{1}{\tan(\theta)}
\]
So, the given equation becomes:
\[
\tan(\theta) + \frac{1}{\tan(\theta)} = 5
\]

### Step 2: Let \( x = \tan(\theta) \)
Substitute \( x \) for \( \tan(\theta) \), and rewrite the equation:
\[
x + \frac{1}{x} = 5
\]
Multiply both sides by \( x \) to eliminate the fraction:
\[
x^2 + 1 = 5x
\]
Rearranging this gives:
\[
x^2 - 5x + 1 = 0
\]
This is a quadratic equation.

### Step 3: Solve the quadratic equation
We can solve the quadratic using the quadratic formula:
\[
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(1)}}{2(1)}
\]
\[
x = \frac{5 \pm \sqrt{25 - 4}}{2}
\]
\[
x = \frac{5 \pm \sqrt{21}}{2}
\]

So, the two possible solutions for \( x \) are:
\[
x = \frac{5 + \sqrt{21}}{2} \quad \text{or} \quad x = \frac{5 - \sqrt{21}}{2}
\]

### Step 4: Find \( \tan^2(\theta) + \cot^2(\theta) \)
We know:
\[
\tan^2(\theta) + \cot^2(\theta) = x^2 + \frac{1}{x^2}
\]
Using the identity:
\[
x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2
\]
Substituting \( x + \frac{1}{x} = 5 \), we get:
\[
x^2 + \frac{1}{x^2} = 5^2 - 2 = 25 - 2 = 23
\]

Thus, \( \tan^2(\theta) + \cot^2(\theta) = 23 \).

Would you like further details on this solution or have any questions?

### Related Questions:
1. What is the value of \( \tan(\theta) \) if \( \tan(\theta) + \cot(\theta) = 7 \)?
2. Can you solve for \( \sin(\theta) \) and \( \cos(\theta) \) given \( \tan(\theta) + \cot(\theta) = 5 \)?
3. What is the sum of the squares of \( \tan(\theta) \) and \( \cot(\theta) \) for other values of \( \theta \)?
4. How do you derive the identity \( \tan^2(\theta) + \cot^2(\theta) = ( \tan(\theta) + \cot(\theta) )^2 - 2 \)?
5. What is the general solution for \( \tan(\theta) + \cot(\theta) = c \), where \( c \) is a constant?

### Tip:
When dealing with trigonometric identities, substituting one trigonometric function in terms of another, such as using \( \cot(\theta) = 1/\tan(\theta) \), is often a helpful strategy for simplification.

If tan(θ) + cot(θ) = 5, what is tan²(θ) + cot²(θ)?

Learn how to solve the trigonometric problem where tan(θ) + cot(θ) = 5, and find the value of tan²(θ) + cot²(θ). This problem involves key trigonometric identities and algebraic manipulation, suitable for students in Grades 10-12.

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