Q. 5

<span lang="EN-US

Answer :

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°.tan 44°.1.

(∵ tan θ =

= 1

Hence, tan 1°. tan 2°.tan 3° …… tan 89° = 1

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