If `"tan"theta+"cot"theta=6`, then find the value of `"tan"^(2)theta+"cot"^(2)theta`.

Options

A. 36
B. 24
C. 26
D. 34

Answer

To solve the problem where we need to find the value of \( \tan^2 \theta + \cot^2 \theta \) given that \( \tan \theta + \cot \theta = 6 \), we can follow these steps:

### Step 1: Square both sides of the equation
We start with the equation:
\[ \tan \theta + \cot \theta = 6 \]
Now, we square both sides:
\[ (\tan \theta + \cot \theta)^2 = 6^2 \]
This gives us:
\[ \tan^2 \theta + 2 \tan \theta \cot \theta + \cot^2 \theta = 36 \]

### Step 2: Use the identity for \( \tan \theta \cot \theta \)
Recall that:
\[ \tan \theta \cot \theta = 1 \]
Thus, we can substitute \( \tan \theta \cot \theta \) in our equation:
\[ \tan^2 \theta + 2(1) + \cot^2 \theta = 36 \]
This simplifies to:
\[ \tan^2 \theta + \cot^2 \theta + 2 = 36 \]

### Step 3: Isolate \( \tan^2 \theta + \cot^2 \theta \)
Now, we can isolate \( \tan^2 \theta + \cot^2 \theta \):
\[ \tan^2 \theta + \cot^2 \theta = 36 - 2 \]
This results in:
\[ \tan^2 \theta + \cot^2 \theta = 34 \]

### Final Answer
Thus, the value of \( \tan^2 \theta + \cot^2 \theta \) is:
\[ \boxed{34} \]