Question

If tanA+cotA=2
Then find the value of tan^5A +cot^5A?

Answer 1

tanA - cotA)^2 
= (tanA + cotA)^2 - 4tanA cotA 
= 2^2 - 4 
=0 
=> tanA = cot A = 1 
=> tan^5 A + cot^5 A 
= 1 + 1 
= 2.

Answer 2

If tan A + cot A = 2 then tan⁵ A + cot⁵ A = 2

Given :

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

To find :

$\displaystyle \sf{ {\tan}^{5} A + {\cot}^{5} A }$

Solution :

Step 1 of 3 :

Write down the given equation

Here it is given that

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

Step 2 of 3 :

Find the value of tan A and cot A

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

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

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

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

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

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

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

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

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

Step 3 of 3 :

Find the value of the expression

$\displaystyle \sf{ {\tan}^{5} A + {\cot}^{5} A }$

$\displaystyle \sf{ = {1}^{5} + {1}^{5} }$

$\sf{ =1 + 1}$

$= 2$

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