Question

If $\tan \left(A\right)+cot\left(A\right)=4$, then ${\tan }^4\left(A\right)+co{t}^4\left(A\right)$ is equal to

• A. $110$
• B. $191$
• C. $80$
• D. $194$

Answer 1

The correct option is D

$194$

Explanation for the correct option:

Step 1: Given formation

$\tan \left(A\right)+cot\left(A\right)=4$

Squaring on both sides,

$\begin{array}{rcl}{\left[\tan \left(A\right)+cot\left(A\right)\right]}^2& =& {\left(4\right)}^2\\ {\tan }^2\left(A\right)+co{t}^2\left(A\right)+2\tan \left(A\right)cot\left(A\right)& =& 16\\ {\tan }^2\left(A\right)+co{t}^2\left(A\right)-2& =& 16\\ {\tan }^2\left(A\right)+co{t}^2\left(A\right)& =& 16-2\\ {\tan }^2\left(A\right)+co{t}^2\left(A\right)& =& 14\end{array}$

Step 2: Squaring again on both sides,

$\begin{array}{rcl}{\left[{\tan }^2\left(A\right)+co{t}^2\left(A\right)\right]}^2& =& {\left(14\right)}^2\\ {\tan }^4\left(A\right)+co{t}^4\left(A\right)+2{\tan }^2\left(A\right)cot\left(A\right)& =& 196\\ {\tan }^4\left(A\right)+co{t}^4\left(A\right)-2& =& 196\\ {\tan }^4\left(A\right)+co{t}^4\left(A\right)& =& 196-2\\ {\tan }^4\left(A\right)+co{t}^4\left(A\right)& =& 194\end{array}$

Hence, the correct option is (D).