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. $\dfrac{1}{2}$
• D. $\dfrac{1}{4}$
• E. $\dfrac{1}{16}$

Answer 1

The Correct Option is C

The points $(\cos \theta_i, \sec \theta_i)$ lie on the boundary of a circle, implying that they satisfy the equation of a circle, i.e., the sum of squares of the coordinates equals 1. Therefore, we can write the relationship as:

$$\cos^2 \theta_i + \sec^2 \theta_i = 1$$

Now, recalling the identity for secant: $\sec \theta = \frac{1}{\cos \theta}$, we substitute this into the equation to get:

$$\cos^2 \theta_i + \left(\frac{1}{\cos \theta_i}\right)^2 = 1$$

This simplifies to:

$$\cos^2 \theta_i + \frac{1}{\cos^2 \theta_i} = 1$$

Multiplying through by $\cos^2 \theta_i$ gives:

$$\cos^4 \theta_i + 1 = \cos^2 \theta_i$$

Rearranging the terms, we find:

$$\cos^4 \theta_i - \cos^2 \theta_i + 1 = 0$$

This equation can be solved numerically or through further algebraic manipulations depending on the properties of the angles $\theta_1, \theta_2, \theta_3, \theta_4$.

Answer: The value of $\cos \theta_1 \cos \theta_2 \cos \theta_3 \cos \theta_4$ is $\frac{1}{16}$, so the correct option is (D).