Question

The value of the expression \(\cos \left(\frac{\pi }{24}\right)\cos \left(\frac{\pi }{12}\right)\cos \left(\frac{\pi }{6}\right)\cos \left(\frac{\pi }{3}\right)\) is:

• A. \(\frac{1}{16}\sin \left(\frac{\pi }{24}\right)\)
• B. \(\frac{\sqrt{3}}{32}\csc \left(\frac{\pi }{24}\right)\)
• C. \(\frac{\sqrt{3}}{16}\)
• D. \(\frac{1}{32}\sec \left(\frac{\pi }{12}\right)\)

Hint:

The angles are in the ratio 1:2:4:8, so use repeated double angle formula.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Put \(\theta=\dfrac{\pi}{24}\). Then the product is \(\cos\theta\cos2\theta\cos4\theta\cos8\theta\).

Step 2: Use the identity
\(\cos\theta\cos2\theta\cos4\theta\cos8\theta=\dfrac{\sin16\theta}{16\sin\theta}\)

Step 3: Substitute
\(16\theta=\dfrac{2\pi}{3}\) and \(\sin\dfrac{2\pi}{3}=\dfrac{\sqrt3}{2}\), so the value is \(\dfrac{\sqrt3}{2\cdot16\sin(\pi/24)}\)

Step 4: Simplify
\(=\dfrac{\sqrt3}{32}\csc\left(\dfrac{\pi}{24}\right)\)

Final Answer:
The value equals root 3 over 32 times cosecant of pi by 24. \[ \boxed{\text{(B) }\dfrac{\sqrt{3}}{32}\csc\left(\dfrac{\pi}{24}\right)} \]