Question

If $\cos x=\tan y,\cot y=\tan z$ and $\cot z=\tan x$, then $\sin x$ is equal to

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

Answer 1

Answer: (D)

$\frac{\sqrt{5}-1}{2}$

Given: $\cos x=\tan y,\cot y=\tan z$ and $\cot z=\tan x$

$\cos x=\tan y$

$\Rightarrow \cos x=\frac{1}{\tan x}$

$\Rightarrow \cos x=\cos z$

$\Rightarrow \cos x=\tan x$

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

$\Rightarrow {\cos }^{2}x=\sin x$

$\Rightarrow 1-{\sin }^{2}x=\sin x$

$\Rightarrow {\sin }^{2}x+\sin x-1=0$

$\therefore \sin x=\frac{-1\pm \sqrt{1-4\times (-1)}}{2\times 1}$

$\Rightarrow \sin x=\frac{\sqrt{5}-1}{2}$ ..$[\because \frac{-1-\sqrt{5}}{2}<-1]$