Question

Find the value of \( \cot 10^\circ \times \cot 30^\circ \times \cot 45^\circ \times \cot 60^\circ \times \cot 80^\circ \)

Hint:

Use trigonometric identities and complementary angle properties to simplify trigonometric products.

Answer 1

Step 1: Use the identity for cotangent.
The cotangent function satisfies the identity: \[ \cot x = \frac{1}{\tan x} \] So the product becomes: \[ \cot 10^\circ \times \cot 30^\circ \times \cot 45^\circ \times \cot 60^\circ \times \cot 80^\circ = \frac{1}{\tan 10^\circ \times \tan 30^\circ \times \tan 45^\circ \times \tan 60^\circ \times \tan 80^\circ} \]
Step 2: Use known values of tangent.
We know the following values: \[ \tan 45^\circ = 1, \quad \tan 30^\circ = \frac{1}{\sqrt{3}}, \quad \tan 60^\circ = \sqrt{3} \] Now, use the identity for complementary angles: \[ \tan(90^\circ - x) = \cot x \] Thus, \( \tan 80^\circ = \cot 10^\circ \). Therefore, the product simplifies to: \[ \frac{1}{\tan 10^\circ \times \tan 30^\circ \times \tan 45^\circ \times \tan 60^\circ \times \tan 80^\circ} \] The final result simplifies to: \[ \boxed{3} \]