Q.142. Addition Formula for tangent function

Question 142.
Prove that tan 10° = (tan20°)(tan30°)(tan40°).

Answer 142.
tan20°tan30°tan40°
= tan30° * tan(30° - 10°) * tan(30° + tan10°)
= tan30° * [(tan30° - tan10°)/(1 + tan30°tan10°)] * [(tan30° + tan10°)/(1 - tan30°tan10°)]
= tan30° * [(tan^2 30° - tan^2 10°)/(1 - tan^2 30° * tan^2 10°)
= tan30° * [(1/3 - tan^2 10°)/(1 - (1/3)tan^2 10°)]
= tan30° * [(1 - 3tan^2 10°)/(3 - tan^2 10°)]
= tan30° * tan10° * [(1 - 3tan^2 10°)/(3tan10° - tan^3 10°)]
... [Multiplying and dividing by tan10°]
= tan30° * tan10° * 1/tan30°
= tan10°.

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