Question

Prove the following:

tan8θ − tan5θ − tan3θ = tan8θ tan5θ tan3θ

Answer 1

∵  8θ = 5θ + 3θ

∴ tan8θ = tan(5θ + 3θ)

∴ tan8θ = `(tan5theta + tan3theta)/(1 - tan5theta . tan3theta)`

∴ tan8θ (1 − tan5θ . tan3θ) = tan5θ + tan3θ

∴ tan 80 – tan 8θ · tan 5θ · tan 3θ = tan 5θ + tan 3θ

∴ tan 8θ – tan 5θ – tan 3θ = tan 8θ · tan 5θ · tan 30