If tan x + cot x = 4,

Given tan x + cot x = 4

Since $\left(\tan ^{2} x+\cot ^{2} x\right)^{2}=\tan ^{4} x+\cot ^{4} x+2 \cot ^{2} x \tan ^{2} x$

i.e $\left(\tan ^{2} x+\cot ^{2} x\right)^{2}=\tan ^{4} x+\cot ^{4} x+2$

i. e $\tan ^{4} x+\cot ^{4} x=\left(\tan ^{2} x+\cot ^{2} x\right)^{2}-2$   ....(1)

also, $(\tan x+\cot x)^{2}=(4)^{2}$

i. e $\tan ^{2} x+\cot ^{2} x+2 \cot x \tan x=16$

$\tan ^{2} x+\cot ^{2} x+2=16$

$\tan ^{2} x+\cot ^{2} x=14$

$\therefore$ equation (1), reduces to

$\tan ^{4} x+\cot ^{4} x=(14)^{2}-2$

i. e $\tan ^{4} x+\cot ^{4} x=194$