Question

lf $\cos x=\tan y,\cos y=\tan z,\cos z=\tan x$ then value of $\sin x$ is

• A. $2\cos {18}^0$
• B. $\cos {18}^0$
• C. $2\sin {18}^0$
• D. $\sin {18}^0$

Answer 1

The correct option is C $2\sin {18}^0$

Given

$\cos x=\tan y\cdots \left(1\right)$

$\cos y=\tan z\cdots \left(2\right)$

$\cos z=\tan x\cdots \left(3\right)$

Let $x=y=z$

$⟹\cos x=\tan x$

$⟹\cos x=\frac{\sin x}{\cos x}$

$⟹\sin x={\cos }^{2}x$

$⟹\sin x=1-{\sin }^{2}x$

$⟹{\sin }^{2}x+\sin x-1=0$

$⟹\sin x=\frac{-1\pm \sqrt{1+4}}{2}$

$⟹\sin x=\frac{-1\pm \sqrt{5}}{4}$
$\because \sin {18}^o=\frac{-1\pm \sqrt{5}}{2}$

$⟹\sin x=2\sin {18}^{\circ }$