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

The Correct Option is C

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) : \(\frac{1}{4}\)