Question

If sin$\theta$ + cosec$\theta$ = 2, thenfind the value of sin${}^2$$\theta$ + cosec${}^2$$\theta$.

Answer 1

Given $\sin \theta +\text{cosec}\theta =2$

$⟹\sin \theta +\frac{1}{\sin \theta }=2$

squaring on both sides

$(\sin \theta +\frac{1}{\sin \theta }{)}^2=4$

$⟹{\sin }^2\theta +\frac{1}{{\sin }^2\theta }+2\left(\sin \theta \right)(\frac{1}{\sin \theta })=4$

$⟹{\sin }^2\theta +\frac{1}{{\sin }^2\theta }+2=4$

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

$⟹{\sin }^2\theta +{\text{cosec}}^2\theta =2$