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Grade: 12
        tan15.tan25.tan35=tanx

find x in degree where x belongs to(0,45)
6 years ago

Answers : (1)

Aman Bansal
592 Points
							

Dear Anil

1. Using the identity: tan x * tan(60° - x) * tan(60° + x) = tan(3x) well have for x = 5°:
tan 5° tan 55° tan 65° = tan 15° = cot 75° = 1/tan 75°, hence
tan 5° tan 55° tan 65° tan 75° = 1 or
tan 5° = 1/(tan 55° tan 65° tan 75°) = cot 55° cot 65° cot 75° =
= tan 35° tan 25° tan 15°, so the answer is t = 5°
Proof of the identity: tan x * tan(60° - x) * tan(60° + x) =
(sin x * sin(60° - x) * sin(60° + x)) / (cos x * cos(60° - x) * cos(60° + x)) =
= (sin x * (cos 2x + 1/2)) / (cos x * (cos 2x - 1/2)) =
= ((1/2)(sin 3x - sin x) + (1/2) sin x) / ((1/2)(cos 3x + cos x) - (1/2) cos x) = tan 3x

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Aman Bansal

Askiitian Expert


6 years ago
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