Question

If tan A + cot A = 2, then the value of tan⁴ A + cot⁴ A =

• A. 2
• B. 1
• C. 4
• D. 5

Answer 1

The Correct Option is A

Given tan A + cot A = 2. Since \(\cot A = \frac{1}{\tan A}\), we can rewrite the given equation as: \(\tan A + \frac{1}{\tan A} = 2\)

Let \(x = \tan A\). 

Then \(x + \frac{1}{x} = 2\). 

Multiplying by \(x\) gives \(x^2 + 1 = 2x\), or \(x^2 - 2x + 1 = 0\). 

This factors as \((x-1)^2 = 0\), so \(x=1\). Thus, \(\tan A = 1\), which implies \(\cot A = \frac{1}{\tan A} = 1\).

Now, tan⁴ A + cot⁴ A = 1⁴ + 1⁴ = 1 + 1 = 2.

Answer: (A) 2

Answer 2

We are given:

$$ \tan A + \cot A = \tan A + \frac{1}{\tan A} = 2. $$

Let $ x = \tan A $. Then:

$$ x + \frac{1}{x} = 2 \implies x^2 - 2x + 1 = 0 \implies (x - 1)^2 = 0 \implies x = 1. $$

Thus, $ \tan A = 1 $ and $ \cot A = \frac{1}{\tan A} = 1 $. 

Now calculate $ \tan^4 A + \cot^4 A $:

$$ \tan^4 A + \cot^4 A = 1^4 + 1^4 = 1 + 1 = 2. $$