If `(tan theta + cot theta)=5` then find the value of `(tan^(2) theta + cot^(2) theta)`. Options are (a) 21 (b) 27 (c) 25 (d) 23.

Options

A. 21
B. 27
C. 25
D. 23

Answer

To solve the problem, we start with the given equation:

\[ \tan \theta + \cot \theta = 5 \]

**Step 1: Square both sides of the equation.**

\[ (\tan \theta + \cot \theta)^2 = 5^2 \]

**Step 2: Expand the left-hand side using the formula \((a + b)^2 = a^2 + b^2 + 2ab\).**

\[ \tan^2 \theta + \cot^2 \theta + 2 \tan \theta \cot \theta = 25 \]

**Step 3: Recall that \(\tan \theta \cot \theta = 1\).**

Substituting this into the equation gives:

\[ \tan^2 \theta + \cot^2 \theta + 2 \cdot 1 = 25 \]

**Step 4: Simplify the equation.**

\[ \tan^2 \theta + \cot^2 \theta + 2 = 25 \]

**Step 5: Isolate \(\tan^2 \theta + \cot^2 \theta\).**

\[ \tan^2 \theta + \cot^2 \theta = 25 - 2 \]

**Step 6: Calculate the final value.**

\[ \tan^2 \theta + \cot^2 \theta = 23 \]

Thus, the value of \(\tan^2 \theta + \cot^2 \theta\) is:

\[ \boxed{23} \]