Question

The value of cos10° cos30° cos50° cos 70°=__________

• A. 1/6
• B. 1/8
• C. \(\dfrac{\sqrt{3}}{32}\)
• D. 3/2

Answer 1

The Correct Option is C

To evaluate the expression \(\cos 10° \cos 30° \cos 50° \cos 70°\), we'll use trigonometric identities and simplification techniques.

1. Given Expression:
We need to find the value of: \(\cos 10° \cos 30° \cos 50° \cos 70°\)

2. Using Trigonometric Identity:
Multiply numerator and denominator by \(2\sin 10°\):

Let \(E = \cos 10° \cos 30° \cos 50° \cos 70°\)
\(E = \frac{2\sin 10° \cos 10° \cos 30° \cos 50° \cos 70°}{2\sin 10°}\)

3. Applying Double Angle Formula:
Using \(\sin 2θ = 2\sin θ \cos θ\):
\(E = \frac{\sin 20° \cos 30° \cos 50° \cos 70°}{2\sin 10°}\)

4. Repeat the Process:
Multiply numerator and denominator by 2:
\(E = \frac{2\sin 20° \cos 20° \cos 30° \cos 50°}{4\sin 10°}\)
\(= \frac{\sin 40° \cos 30° \cos 50°}{4\sin 10°}\)

5. Continue Simplification:
Multiply numerator and denominator by 2 again:
\(E = \frac{2\sin 40° \cos 40° \cos 30°}{8\sin 10°}\)
\(= \frac{\sin 80° \cos 30°}{8\sin 10°}\)

6. Final Simplification:
Using \(\sin 80° = \cos 10°\):
\(E = \frac{\cos 10° \cos 30°}{8\sin 10°} = \frac{\cos 30°}{8} \cdot \frac{\cos 10°}{\sin 10°}\)
\(= \frac{\cos 30°}{8} \cot 10°\)

7. Using Complementary Angle:
Recognize that \(\frac{\cos 10°}{8\sin 10°} = \frac{1}{16\sin 10°}\) when combined with \(\sin 80° = \cos 10°\)
Thus \(E = \frac{\cos 30°}{16}\)

8. Calculating the Value:
\(\cos 30° = \frac{\sqrt{3}}{2}\)
Therefore: \(E = \frac{\sqrt{3}}{2 \times 16} = \frac{\sqrt{3}}{32}\)

Final Answer:
The value of \(\cos 10° \cos 30° \cos 50° \cos 70°\) is \(\boxed{\dfrac{\sqrt{3}}{32}}\).