Question

If $\sec \theta +\tan \theta =2+\sqrt{5}$, then the value of $\sin \theta$ is

• A. $\frac{2}{\sqrt{3}}$
• B. $\frac{1}{\sqrt{5}}$
• C. $\frac{\sqrt{3}}{2}$
• D. $\frac{4}{5}$

Answer 1

The correct option is A $\frac{2}{\sqrt{3}}$

$\sec \theta +\tan \theta =2+\sqrt{5}$
$\sec \theta -\tan \theta =\frac{1}{2+\sqrt{5}}=\frac{1}{\sqrt{5}+2}\times \frac{\sqrt{5}-2}{\sqrt{5}-2}=\sqrt{5}-2=\sqrt{5}-2$
$\sec \theta -\tan \theta =\sqrt{5}-2\Rightarrow 2\sec \theta =2\sqrt{5}$
$\sec \theta =\sqrt{5}\Rightarrow \cos \theta =\frac{1}{\sqrt{5}}$
${\sin }^2\theta =1-{\cos }^2\theta =1-{\left(\frac{1}{\sqrt{5}}\right)}^2=1-\frac{1}{5}=\frac{4}{5}\Rightarrow \sin \theta =\frac{2}{\sqrt{5}}$.