If tanθ + cotθ = 2\(\sqrt {5}\), then find tan³θ - cot³θ.

76

Formula → x + \(\frac{1}{x}\) = y, then x - \(\frac{1}{x}\) = \(\sqrt {y^2 - 4}\) and tanθ = \(\frac{1}{cotθ}\) 

⇒ tanθ + cotθ = 2\(\sqrt {5}\)

⇒ tanθ - cotθ = \(\sqrt {(2\sqrt {5})^2 - 4}\)

⇒ tanθ - cotθ = 4

Now, 

tan³θ - \(\frac{1}{tan^3θ}\) = (4)³ + 3 × 4

⇒ tan³θ - cot³θ = 64 + 12 = 76