Question

Fill in the blanks :
The least value of ${tan}^2\theta +{cot}^2\theta$ is ___________.

Answer 1

${\tan }^2\theta +{\cot }^2\theta$

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

$=\frac{{\sin }^4\theta +{\cos }^4\theta }{{\sin }^2\theta {\cos }^2\theta }$

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

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

For least value, ${\sin }^2\theta {\cos }^2\theta$ should have max value.

Both are max at $\theta ={45}^{\circ }$

$=\frac{1}{\frac{1}{2}\times \frac{1}{2}}-2$

$=4-2=2$

$\therefore$ least value$=2$