Question

Prove the following:

tan8θ − tan5θ − tan3θ = tan8θ · tan5θ · tan3θ

Answer 1

∵  8θ = 5θ + 3θ

∴ tan8θ = tan(5θ + 3θ)     ...`[tan ("A + B") = (tan "A" + tan "B")/(1 - tan "A" * tan "B")]`

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

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

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

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