Question

If $4{\sin }^2\theta -1=0$ and $\theta$ is less than ${90}^o$, find the value of $\theta$ and the value of ${\cos }^2\theta +{\tan }^2\theta$.

Answer 1

Given that,

$4{\sin }^2\theta -1=0$

or, ${\sin }^2\theta =\frac{1}{4}$

or. $\sin \theta =\sqrt{\frac{1}{4}}=\frac{1}{2}$ $(\theta <{90}^{\circ })$

So, $\theta ={30}^{\circ }$

Now,

${\cos }^2\theta +{\tan }^2\theta ={\cos }^2{30}^{\circ }+{\tan }^2{30}^{\circ }$

$=(\frac{\sqrt{3}}{2}{)}^2+(\frac{1}{\sqrt{3}}{)}^2$

$=\frac{3}{4}+\frac{1}{3}$

$=\frac{13}{12}$