Question

Prove that ${(\sin \theta +\csc \theta )}^2+{(\cos \theta +\sec \theta )}^2\ge 8.$

Answer 1

Applying $A.M\ge G.M$ on $\sin \theta ,\csc \theta$
$\frac{\sin \theta +\csc \theta }{2}\ge \sqrt{\sin \theta \csc \theta }\Rightarrow {(\sin \theta +\csc \theta )}^2\ge 4$ ...(1)
Similarly ${(\cos \theta +\sec \theta )}^2\ge 4$ ...(2)
From (1) and (2)
${(\sin \theta +\csc \theta )}^2+{(\cos \theta +\sec \theta )}^2\ge 8$