Question

If \( \tan A + \cot A = 2 \), then the value of \( \tan^2 A + \cot^2 A \) is:

• A. 5
• B. 4
• C. 6
• D. 2

Hint:

Tip: If you see a sum like \( \tan A + \cot A \), try squaring it to get expressions like \( \tan^2 A + \cot^2 A \).

Answer 1

The Correct Option is D

Given:

\(\tan A + \cot A = 2\)

We need to find the value of \(\tan^2 A + \cot^2 A\).

First, recall that \(\cot A = \frac{1}{\tan A}\). Substitute in the given equation:

\(\tan A + \frac{1}{\tan A} = 2\)

Let \(x = \tan A\). Then we have:

\(x + \frac{1}{x} = 2\)

Multiply through by \(x\) to eliminate the fraction:

\(x^2 + 1 = 2x\)

Rearrange to form a quadratic equation:

\(x^2 - 2x + 1 = 0\)

Observe that this is a perfect square:

\((x-1)^2 = 0\)

Thus, \(x = 1\), which means \(\tan A = 1\).

Now, find \(\tan^2 A + \cot^2 A\):

\(\tan^2 A = 1^2 = 1\)

\(\cot A = \frac{1}{\tan A} = 1\), so \(\cot^2 A = 1^2 = 1\)

Therefore, \(\tan^2 A + \cot^2 A = 1 + 1 = 2\).

The value of \(\tan^2 A + \cot^2 A\) is 2.

Answer 2

Step 1: Let \( x = \tan A \), then \( \cot A = \frac{1}{x} \) 
Given: \( x + \frac{1}{x} = 2 \)

Step 2: Square both sides. 
\[ \left( x + \frac{1}{x} \right)^2 = 4 \Rightarrow x^2 + \frac{1}{x^2} + 2 = 4 \Rightarrow x^2 + \frac{1}{x^2} = 2 \]

Step 3: Recognize the expression. 
Since \( \tan^2 A + \cot^2 A = x^2 + \frac{1}{x^2} \), the value is \( \boxed{2} \) \[ \left( x + \frac{1}{x} \right)^2 = x^2 + \frac{1}{x^2} + 2 = 4 \Rightarrow x^2 + \frac{1}{x^2} = 2 \]
\(\Rightarrow \boxed{\text{Correct Answer is (D) 2}} \)