Question

If $\frac{\sin \alpha}{\sin \beta}=\frac{\sqrt{3}}{2}$ and $\frac{\cos \alpha}{\cos \beta}=\frac{\sqrt{5}}{2}, 0 < \alpha < \beta < \frac{\pi}{2}$, then which of the following is false

• A. $\cot \beta-1$
• B. $\tan \alpha-\frac{\sqrt{3}}{\sqrt{5}}$
• C. $B=\frac{\pi}{4}$
• D. $B =\frac{\pi}{3}$

Answer 1

Answer: (C)

$\sin \alpha =\frac{\sqrt{3}}{2} \sin \beta$
$\Rightarrow \cos \alpha =\frac{\sqrt{5}}{2} \cos \beta$
Squaring and adding, we get
$\Rightarrow 1=\frac{3}{4} \sin ^{2} \beta=\frac{5}{4}\left(1-\sin ^{2} \beta\right)$
$\frac{1}{2} \sin ^{2} \beta=\frac{1}{4}$
$\Rightarrow \sin ^{2} \beta=\frac{1}{2}$
$\beta=\frac{\pi}{4}$
$\Rightarrow \tan \alpha=\sqrt{\frac{3}{5}}$