Question

The value of the product $\cot 10^\circ \cot 20^\circ \cot 30^\circ \cot 45^\circ \cot 60^\circ \cot 70^\circ \cot 80^\circ$ is equal to

• A. $\sqrt{3}$
• B. $\frac{\sqrt{3}}{3}$
• C. $3$
• D. $3\sqrt{3}$
• E. $1$

Hint:

Whenever you see a product of trigonometric ratios with angles summing to $90^\circ$, look for complementary identity pairings. Most such products simplify to 1 or $\sqrt{3}$.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
The complementary angle identity for cotangent is $\cot \theta = \tan(90^\circ - \theta)$. Also, $\cot \theta \cdot \tan \theta = 1$.

Step 2: Detailed Explanation:
1. Identify pairs of complementary angles in the product:
$\cot 80^\circ = \tan(90^\circ - 80^\circ) = \tan 10^\circ$.
$\cot 70^\circ = \tan(90^\circ - 70^\circ) = \tan 20^\circ$.
$\cot 60^\circ = \tan(90^\circ - 60^\circ) = \tan 30^\circ$.
2. Group the terms:
\[ P = (\cot 10^\circ \cdot \cot 80^\circ) \cdot (\cot 20^\circ \cdot \cot 70^\circ) \cdot (\cot 30^\circ \cdot \cot 60^\circ) \cdot \cot 45^\circ \]
3. Substitute the tan equivalents:
\[ P = (\cot 10^\circ \cdot \tan 10^\circ) \cdot (\cot 20^\circ \cdot \tan 20^\circ) \cdot (\cot 30^\circ \cdot \tan 30^\circ) \cdot \cot 45^\circ \]
4. Since $\cot \theta \cdot \tan \theta = 1$ and $\cot 45^\circ = 1$:
\[ P = 1 \cdot 1 \cdot 1 \cdot 1 = 1 \]

Step 3: Final Answer:
The product is equal to 1.