Question

If tan A + cot A = 4, then tan⁴ A + cot⁴ A is equal to

Options

• 110

• 191

• 80

• 194

Answer 1

194

We have: 

\[\tan A + \cot A = 4\]

Squaring both the sides:

\[ \left( \tan A + \cot A \right)^2 = 4^2 \]

\[ \Rightarrow \tan^2 A + \cot^2 A + 2 \left( \tan A \right)\left( \cot A \right) = 16\]

\[ \Rightarrow \tan^2 A + \cot^2 A + 2 = 16\]

\[ \Rightarrow \tan^2 A + \cot^2 A = 14\]

Squaring both the sides again: 

\[ \left( \tan^2 A + \cot^2 A \right)^2 = {14}^2 \]

\[ \Rightarrow \tan^4 A + \cot^4 A + 2 \left( \tan^2 A \right)\left( \cot^2 A \right) = 196\]

\[ \Rightarrow \tan^4 A + \cot^4 A + 2 = 196\]

\[ \Rightarrow \tan^4 A + \cot^4 A = 194\]