If tan A + cot A = 4,

Question:

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

(a) 110

(b) 191

(c) 80

(d) 194

Solution:

(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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