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:
We have a product of cotangent functions for complementary angles.
We know that \(\cot(90^\circ - \theta) = \tan \theta\) and \(\cot \theta \cdot \tan \theta = 1\).
Step 2: Key Formula or Approach:
Pair up terms whose angles sum to \(90^\circ\).
Evaluate \(\cot 45^\circ\), \(\cot 30^\circ\), and \(\cot 60^\circ\) directly.
Step 3: Detailed Explanation:
Rewrite the product by grouping complementary angles:
\[ (\cot 10^\circ \cdot \cot 80^\circ) \cdot (\cot 20^\circ \cdot \cot 70^\circ) \cdot \cot 30^\circ \cdot \cot 60^\circ \cdot \cot 45^\circ \] Convert the higher angles using \(\cot(90^\circ - \theta) = \tan \theta\):
\(\cot 80^\circ = \tan 10^\circ\)
\(\cot 70^\circ = \tan 20^\circ\)
Now substitute them back:
\[ (\cot 10^\circ \cdot \tan 10^\circ) \cdot (\cot 20^\circ \cdot \tan 20^\circ) \cdot \cot 30^\circ \cdot \cot 60^\circ \cdot \cot 45^\circ \] Since \(\cot \theta \cdot \tan \theta = 1\), the first two pairs become 1.
\[ 1 \cdot 1 \cdot \cot 30^\circ \cdot \cot 60^\circ \cdot \cot 45^\circ \] Now use standard values:
\(\cot 30^\circ = \sqrt{3}\)
\(\cot 60^\circ = \frac{1}{\sqrt{3}}\)
\(\cot 45^\circ = 1\)
Substitute these values:
\[ 1 \cdot 1 \cdot \sqrt{3} \cdot \frac{1}{\sqrt{3}} \cdot 1 = 1 \] Step 4: Final Answer:
The product evaluates to 1.