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:

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

Determine the value of \(\tan^2 A + \cot^2 A\).

Using the identity \(\cot A = \frac{1}{\tan A}\), substitute into the given equation:

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

Let \(x = \tan A\). The equation becomes:

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

Multiply by \(x\) to clear the denominator:

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

Rearrange into a quadratic equation:

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

This equation is a perfect square trinomial:

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

Solving for \(x\), we get \(x = 1\), which implies \(\tan A = 1\).

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

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

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

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

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