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Question
The value of (tan1° tan2° tan3° ... tan89°) is ______.
Options
0
1
2
`1/2`
Solution
The value of (tan1° tan2° tan3° ... tan89°) is 1.
Explanation:
tan 1°.tan 2°.tan 3° ...... tan 89°
= tan 1°.tan 2°.tan 3°...tan 43°.tan 44°.tan 45°.tan 46°.tan 47°...tan 87°.tan 88°.tan 89°
Since, tan 45° = 1,
= tan 1°.tan 2°.tan 3°...tan 43°.tan 44°.1.tan 46°.tan 47°...tan 87°.tan 88°.tan 89°
= tan 1°.tan 2°.tan 3°…tan 43°.tan 44°.1.tan(90° – 44°).tan(90° – 43°) ...tan(90° – 3°).tan(90° – 2°).tan(90° – 1°)
Since, tan(90° – θ) = cot θ,
= tan 1°.tan 2°.tan 3°...tan 43°.tan 44°.1.cot 44°.cot 43°...cot 3°.cot 2°.cot 1°
Since, tan θ = `(1/cot θ)`
= `tan1^circ * tan2^circ * tan3°...tan43^circ * tan44^circ * 1 * (1/tan 44^circ)`. `(1/tan 43^circ) ... (1/tan 3^circ) * (1/tan 2^circ) * (1/tan 1^circ)`
= `(tan 1^circ xx 1/tan1^circ) * (tan 2^circ xx 1/tan 2^circ) ... (tan 44^circ xx 1/tan 44^circ)`
= 1
Hence, tan 1°.tan 2°.tan 3° ...... tan 89° = 1
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