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

OR

If A, B, C, are the interior angles of a triangle ABC, prove that

tan 1°. tan 2°.tan 3° …… tan 89°

= tan1°.tan 2°.tan 3°…tan 43°.tan 44°.tan 45°.tan 46°.tan 47°…tan 87°.tan 88°.tan 89°

= tan1°.tan 2°.tan 3°…tan 43°.tan 44°.1.tan 46°.tan 47°…tan 87°.tan 88°.tan 89° (∵ tan 45° = 1)

= tan1°.tan 2°.tan 3°…tan 43°.tan 44°.1.tan(90° – 44°).tan(90° – 43°)…tan(90° – 3°). tan(90° – 2°).tan(90° – 1°)

= tan1°.tan 2°.tan 3°…tan 43°.tan 44°.1.cot 44°.cot 43°…cot 3°.cot 2°.cot 1° (∵ tan(90° – θ)=cot θ)

= tan1°.tan 2°.tan 3°…tan 43°. tan44°.1.

(∵ tan θ =

= 1

OR

Since, A, B, C, are the interior angles of a triangle ABC.

Therefore,

Hence proved.