If + =4, then ^4A+ ^4A is equal to: - Vidyadip

MCQ

If $\tan\text{A}+\cot\text{A}=4,$ then $\tan^4\text{A}+\cot^4\text{A}$ is equal to:

A
110
B
191
C
80
✓
194

✓

Answer

Correct option: D.

194

We have:
$\tan\text{A}+\cot\text{A}=4$
squaring both the sides:
$(\tan\text{A}+\cot\text{A})^2=4^2$
$\Rightarrow\tan^2\text{A}+\cot^2\text{A}+2(\tan\text{A})(\cot\text{A})=16$
$\Rightarrow\tan^2\text{A}+\cot^2\text{A}+2=16$
$\Rightarrow\tan^2\text{A}+\cot\text{A}=14$
squaring both the sides again:
$(\tan^2\text{A}+\cot^2\text{A})^2=14^2$
$\tan^4\text{A}+\cot^4\text{A}+2(\tan^2\text{A})(\cot^2\text{A})=196$
$\Rightarrow\tan^4\text{A}+\cot^4\text{A}+2=196$
$\Rightarrow\tan^4\text{A}+\cot^4\text{A}=194$

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