Question

For $i=1,2,3,4$, suppose the points$(\cosθi,\secθi)$ lie on the boundary of a circle,where $θi∈[0,\frac{π}{6})$ are distinct. Then $\cosθ_1\ \cosθ_2\ \cosθ_3\ \cosθ_4$ equals

• A. $\dfrac{1}{2}$
• B. $\dfrac{1}{8}$
• C. $\dfrac{1}{4}$
• D. $\dfrac{1}{2}$
• E. $\dfrac{1}{16}$

Answer 1

Answer: (C)

$\dfrac{1}{4}$

Answer 2

Given that:
For $i=1,2,3,4$,suppose the points $(\cosθi,\secθi)$ lie on the boundary of a circle [ where$θi∈[0,\frac{π}{6})$ ]
let general point $(\cos θ,\sec θ)$ and radius is $1$.
we can write,
$i^2.\cos^2θ + i^2\sec^2θ = 1$
$⇒-\cos^2θ - \dfrac{1}{ \cos^2θ} = 1$

$⇒-\cos^2θ − \cos^2θ - 1 = 0$

$$$⇒-2\cos^2θ - 1 = 0$

$⇒\cos^2θ =\dfrac{1}{2}$

$⇒\cosθ =\dfrac{1}{√2}$
Therefore product roots: $\cos θ_1\ \cos θ_2\ \cos θ_3\ \cos θ_4 = \dfrac{1}{4}$
So, the correct option is (C) : $\frac{1}{4}$