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Question
If tan A + cot A = 4, then tan4 A + cot4 A is equal to
Options
110
191
80
194
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Solution
194
We have:
\[\tan A + \cot A = 4\]
Squaring both the sides:
\[ \left( \tan A + \cot A \right)^2 = 4^2 \]
\[ \Rightarrow \tan^2 A + \cot^2 A + 2 \left( \tan A \right)\left( \cot A \right) = 16\]
\[ \Rightarrow \tan^2 A + \cot^2 A + 2 = 16\]
\[ \Rightarrow \tan^2 A + \cot^2 A = 14\]
Squaring both the sides again:
\[ \left( \tan^2 A + \cot^2 A \right)^2 = {14}^2 \]
\[ \Rightarrow \tan^4 A + \cot^4 A + 2 \left( \tan^2 A \right)\left( \cot^2 A \right) = 196\]
\[ \Rightarrow \tan^4 A + \cot^4 A + 2 = 196\]
\[ \Rightarrow \tan^4 A + \cot^4 A = 194\]
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