Question

If $\cos x=\tan y,coty=\tan z$ and $cotz=\tan x$, 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

The correct option is D

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

Explanation for the correct option:

Step 1. Find the value of $\sin x$:

Given, $\cos x=\tan y$ ….(1)

$coty=\tan z$

$\Rightarrow$ $\tan y=cotz$ ….(2)

$cotz=\tan x$ ….(3)

Step 2. Equate equation (2) and (3),

$\tan y=\tan x$

$\Rightarrow$ $\cos x=\tan x$

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

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

$\Rightarrow$ $1–{\sin }^{2}x=\sin x$

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

It is a quadratic equation in $\sin x$.

Step 3. By Using quadratic formula, we get

$\begin{array}{rcl}\sin x& =& \frac{[-1\pm \sqrt{(1+4)}]}{2}\\ & =& \frac{-1\pm \sqrt{5}}{2}\end{array}$

$\sin x=\frac{(-1+\sqrt{5})}{2},\frac{(-1–\sqrt{5})}{2}$

$\mathbf{\therefore }\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{x}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{(}\sqrt{\mathbf{5}}\mathbf{}\mathbf{–}\mathbf{}\mathbf{1}\mathbf{)}}{\mathbf{2}}$ $\left[\mathbf{\because }\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{x}\mathbf{}\mathbf{\in }\mathbf{}\left[-1,1\right]\right]$

Hence, Option ‘D’ is Correct.