Question

Evaluate :
${(sin\theta +cosec\theta )}^2+{(cos\theta +sec\theta )}^2$

Answer 1

given, ${(\sin \theta +\csc \theta )}^2+{(\cos \theta +\sec \theta )}^2$

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

$=({\sin }^2\theta +{\cos }^2\theta )+({\sec }^2\theta +{\csc }^2\theta )+2+2$ using reciprocal relations of trigonometry.

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

$=5+\frac{{\sin }^2\theta +{\cos }^2\theta }{{\sin }^2\theta {\cos }^2\theta }$

$=5+\frac{1}{{\sin }^2\theta {\cos }^2\theta }$

$=5+{\csc }^2\theta {\sec }^2\theta$