Question

If $\log \frac{\cos x}{\sin x}+\log \frac{\tan x}{\cot x}+\log \frac{\sec x}{\cos ecx}=0$.Find $\frac{\sin x}{\sin \left(\frac{\pi }{10}\right)}$

Answer 1

We know that $\log a+\log b=\log ab$. using this property:

$\log \frac{\cos x}{\sin x}+\log \frac{\tan x}{\cot x}+\log \frac{\sec x}{\cos ecx}=0$

$\Rightarrow \log (\frac{\cos x}{\sin x}\times \frac{\tan x}{\cot x}\times \frac{\sec x}{\cos ecx})=0$

$\Rightarrow \log {\left(\frac{\sin x}{\cos x}\right)}^2=0$

$\Rightarrow \sin x=\cos x\Rightarrow x=\frac{\pi }{4}$

So, $\sin x=\frac{1}{\sqrt{2}}$

Hence,

$\frac{\sin x}{\sin \frac{\pi }{10}}=\frac{1\times 4}{\sqrt{2}(\sqrt{5}-1)}=\frac{2\sqrt{2}}{\sqrt{5}-1}$