If sinα + cosecα = tan (\(\frac{\pi}{3}\))

Find the value of (sin³α + cosec³α)

0

Formula I → [sinα = \(\frac{1}{cosecα}\)]

Formula II → [If x + \(\frac{1}{x}\) = y and x³ + \(\frac{1}{x^3}\) = y³ - 3y]

So,

⇒sinα + \(\frac{1}{sinα}\) = tan \(\frac{180°}{3}\) = tan60°

⇒ sinα + \(\frac{1}{sinα}\) = \(\sqrt {3}\)

Cubing both sides,

⇒ sin³α + \(\frac{1}{sin^3α}\) = (\(\sqrt {3}\))³ - 3\(\sqrt {3}\)

⇒ sin³α + cosec³α = 0