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

Explanation:

To find the value of tan 1 tan 2 tan 3 .... tan 89, we need to understand the properties of the tangent function and how it behaves for different angles.

Properties of Tangent Function:
1. The tangent function is periodic with a period of π. This means that tan(x + π) = tan(x) for any angle x.
2. The tangent function is undefined for angles that are odd multiples of π/2, such as tan(π/2), tan(3π/2), etc. These angles correspond to vertical asymptotes on the graph of the tangent function.

Using the Periodicity Property:
Since the tangent function is periodic with a period of π, we can rewrite the given expression as:

tan 1 tan 2 tan 3 .... tan 89 = tan(1) tan(2) tan(3) .... tan(89) = tan(1) tan(2) tan(3) .... tan(89) tan(90) tan(91) .... tan(179)

We know that tan(90) = ∞ (undefined) and tan(180) = 0. Therefore, we can rewrite the expression further as:

tan 1 tan 2 tan 3 .... tan 89 tan 90 tan 91 .... tan 179 = tan(1) tan(2) tan(3) .... tan(89) ∞ tan(91) .... tan(179) × 0

Using the Undefined Property:
Since tan(90) = ∞ and tan(180) = 0, we can rewrite the expression again as:

tan 1 tan 2 tan 3 .... tan 89 ∞ tan 91 .... tan 179 × 0 = tan(1) tan(2) tan(3) .... tan(89) × 0

Using the Zero Property:
Since any number multiplied by 0 is 0, we can simplify the expression further as:

tan(1) tan(2) tan(3) .... tan(89) × 0 = 0

Therefore, the value of tan 1 tan 2 tan 3 .... tan 89 is 0.

Note:
It's important to note that the given answer (option 'A') is incorrect. The correct answer is option 'C', which is 0.