If $ sin α + cosec α = tan\frac{\pi}{3}$, then the value of $(sin^3 α +cosec^3α)$ is equal to :
Answer & explanation
Correct answer: option 3
sin α + cosec α = tan \(\frac{π}{3}\)
{ \(\frac{π}{3}\) = \(\frac{180º}{3}\) = 60º & tan60º = \(\sqrt {3 }\) }
sin α + cosec α = tan60º
sin α + cosec α = \(\sqrt {3 }\)
sin α + \(\frac{1}{sin α}\) = \(\sqrt {3 }\)
We know,
If x + \(\frac{1}{x}\) = a
then x³ + \(\frac{1}{x³}\) = a³ - 3a
Now,
sin³ α + \(\frac{1}{sin³ α}\) = (\(\sqrt {3 }\))³ - 3(\(\sqrt {3 }\))
= 3(\(\sqrt {3 }\)) - 3(\(\sqrt {3 }\))
= 0