Question

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

• A. $\frac{13}{12}$
• B. $\frac{12}{13}$
• C. $\frac{7}{16}$
• D. $\frac{16}{7}$

Answer 1

The correct option is A $\frac{13}{12}$

$4{\sin }^2\theta -1=0$
$4{\sin }^2\theta =1$
${\sin }^2\theta =\frac{1}{4}$
$\sin \theta =\frac{1}{2}(\sin 30°=\frac{1}{2})$
$\sin \theta =\sin 30°$
$\theta =30°$
${\cos }^2\theta +{\tan }^2\theta$
Putting the value $\theta =30°$
${\cos }^2\theta +{\tan }^2\theta ={\cos }^{2}30°+{\tan }^{2}30°$
$=(\frac{\sqrt{3}}{2}{)}^2+(\frac{1}{\sqrt{3}}{)}^2=\frac{3}{4}+\frac{1}{3}$
$=\frac{9+4}{12}=\frac{13}{12}$
So, Option A is the correct answer.