If $$\tan A + \cot A = \sqrt 5$$, What is the value of $$\tan^3 A + \cot ^3 A$$?

$$cotA = \frac{1}{tanA}$$

let $$tanA = x, then  cotA = \frac{1}{x}$$

Given, x + $$\frac{1}{x} = \sqrt 5$$.............................(1)

Cubing on both sides, we get

$$(x + \frac{1}{x})^3 = \sqrt 5^3$$

$$\Rightarrow x^3 + \frac{1}{x}^3 + 3\times x\times \frac{1}{x}\times (x + \frac{1}{x}) = 5\sqrt 5$$

$$\Rightarrow x^3 + \frac{1}{x}^3 + 3\times \sqrt 5 = 5\sqrt 5$$

$$\Rightarrow x^3 + \frac{1}{x}^3 = 2\sqrt 5$$.

$$\Rightarrow tanA^3 + cotA^3 = 2\sqrt 5$$.