# tan15.tan25.tan35=tanx find x in degree where x belongs to(0,45)

Aman Bansal
592 Points
10 years ago

Dear Anil

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

Aman Bansal