Question

\[ \text{If } \tan A+\cot A=4, \]

\[ \text{then } \tan^4A+\cot^4A \text{ is equal to} \]

(a) \(110\)
(b) \(191\)
(c) \(80\)
(d) \(194\)

Solution

Since

\[ \tan A\cdot \cot A=1 \]

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

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

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

Now,

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

\[ 14^2 = \tan^4A+\cot^4A+2 \]

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

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

Answer

\[ \boxed{194} \]

Correct Option: (d)