Question

If tan A + cot A =√5 then what is the value of tan^3 A + cot^3 A

Answer 1

It is given that the value of tanA + cotA is √5.
= > tanA + cotA = √5

Cube on both sides of the equation given above,

= > ( tanA + cotA )³ = ( √5 )³


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From the properties on expansion, we know : -
( a + b )³ = a³ + b³ + 3ab( a + b )
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= > ( tanA )³ + ( cotA )³ + 3( tanA x cotA )( tanA + cotA ) = √5 x √5 x √5


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From the properties of trigonometry,
cotA = 1 / tanA

Given, tanA + cotA = √5
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= > tan³A + cot³A + 3( tanA x 1 / tanA )( √5 ) = 5√5

= > tan³A + cot³A + 3√5 = 5√5

= > tan³A + cot³A = 5√5 - 3√5

= > tan³A + cot³A = 2√5 or √( 5 x 2² ) = √20



Hence,
The numeric value of tan³A + cot³A is 2√5 or √20.
$\:$

Answer 2

Hello friends!!

Here is your answer :

Please see the attached file..

${\tan}^{3} A \: + { \cot}^{3} A \: = 2 \sqrt{5}$


Hope it helps you ..

^_^