Question

The value of tan 1° tan 2° tan 3° ………. tan 89° is

• A. 0
• B. 1
• C. \(\frac{1}{2}\)
• D. -1

Answer 1

The Correct Option is B

The given problem asks for the value of:
\(\tan 1^\circ \times \tan 2^\circ \times \tan 3^\circ \times \dots \times \tan 89^\circ\)
We can use the key trigonometric identity:
\[ \tan(90^\circ - x) = \cot x \]
So, for each pair \( \tan x^\circ \) and \( \tan(90^\circ - x^\circ) = \cot x^\circ \), their product is:
\[ \tan x^\circ \times \tan(90^\circ - x^\circ) = \tan x^\circ \times \cot x^\circ = 1 \]
Therefore, the product of tangents from \( 1^\circ \) to \( 89^\circ \) can be paired as:
\[ (\tan 1^\circ \times \tan 89^\circ), (\tan 2^\circ \times \tan 88^\circ), \dots \]
Each pair gives a product of 1, and since the number of terms is odd (with \( \tan 45^\circ \) in the middle), the value of the entire product is: \[ 1 \]The correct answer is (B) : 1.