Question

Prove that ${(\sin \theta +\sec \theta )}^2+{(\cos \theta +\csc \theta )}^2={(1+\sec \theta \csc \theta )}^2$

Answer 1

L.H.S$={(\sin \theta +\sec \theta )}^2+{(\cos \theta +\csc \theta )}^2$

$={(\frac{1}{\csc \theta }+\sec \theta )}^2+{(\frac{1}{\sec \theta }+\csc \theta )}^2$

$={\left(\frac{1+\sec \theta \csc \theta }{\csc \theta }\right)}^2+{\left(\frac{1+\sec \theta \csc \theta }{\sec \theta }\right)}^2$

$={(1+\sec \theta \csc \theta )}^2(\frac{1}{{\csc }^2\theta }+\frac{1}{{\sec }^2\theta })$

$={(1+\sec \theta \csc \theta )}^2({\sin }^2\theta +{\cos }^2\theta )$

$={(1+\sec \theta \csc \theta )}^2$