Question

Prove that sec²∅+cosec²∅≥4.
Explain the steps too.

Answer 1

Applying $A.M \ge H.M$ on ${sin}^{2}\phi$ and ${cos}^{2}\phi,\\\Rightarrow\frac{{sin}^{2}\phi+{cos}^{2}\phi}{2}\ge\frac{2}{\frac{1}{{sin}^{2}\phi}+\frac{1}{{cos}^{2}\phi}}\\\Rightarrow\frac{1}{2}\ge\frac{2}{{cosec}^{2}\phi+{sec}^{2}\phi}\\\Rightarrow{cosec}^{2}\phi+{sec}^{2}\phi\ge4\\Q.E.D$