If tan A + cot A = 4, then tan4 A + cot4 A is equal to
(a) 110
(b) 191
(c) 80
(d) 194
(d) 194
We have:
$\tan \mathrm{A}+\cot \mathrm{A}=4$
Squaring both the sides:
$(\tan \mathrm{A}+\cot \mathrm{A})^{2}=4^{2}$
$\Rightarrow \tan ^{2} \mathrm{~A}+\cot ^{2} \mathrm{~A}+2(\tan \mathrm{A})(\cot \mathrm{A})=16$
$\Rightarrow \tan ^{2} \mathrm{~A}+\cot ^{2} \mathrm{~A}+2=16$
$\Rightarrow \tan ^{2} \mathrm{~A}+\cot ^{2} \mathrm{~A}=14$
Squaring both the sides again:
$\left(\tan ^{2} \mathrm{~A}+\cot ^{2} \mathrm{~A}\right)^{2}=14^{2}$
$\Rightarrow \tan ^{4} \mathrm{~A}+\cot ^{4} \mathrm{~A}+2\left(\tan ^{2} \mathrm{~A}\right)\left(\cot ^{2} \mathrm{~A}\right)=196$
$\Rightarrow \tan ^{4} \mathrm{~A}+\cot ^{4} \mathrm{~A}+2=196$
$\Rightarrow \tan ^{4} \mathrm{~A}+\cot ^{4} \mathrm{~A}=194$
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