Question

if tan a + cot a=2, then tan^20a+cot^20a=​

Answer 1

SOLUTION

GIVEN

$\displaystyle \sf{ \tan A + \cot A = 2}$

TO DETERMINE

$\displaystyle \sf{ {\tan}^{20} A + {\cot}^{20} A }$

EVALUATION

Here it is given that

$\displaystyle \sf{ \tan A + \cot A = 2}$

$\displaystyle \sf{ \implies \: \tan A + \frac{1}{\tan A} = 2}$

$\displaystyle \sf{ \implies \: \frac{{\tan}^{2} A + 1}{\tan A} = 2}$

$\displaystyle \sf{ \implies \: {\tan}^{2} A + 1 = 2\tan A}$

$\displaystyle \sf{ \implies \: {\tan}^{2} A + 1 - 2\tan A = 0}$

$\displaystyle \sf{ \implies \: {(\tan A - 1)}^{2} = 0}$

$\displaystyle \sf{ \implies \: (\tan A - 1) = 0}$

$\displaystyle \sf{ \implies \: \tan A = 1}$

$\displaystyle \sf{ \implies \: \cot A = 1}$

This gives

$\displaystyle \sf{ {\tan}^{20} A + {\cot}^{20} A }$

$\displaystyle \sf{ = {1}^{20} + {1}^{20} }$

$\displaystyle \sf{ = 1 + 1 }$

$\displaystyle \sf{ = 2}$

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