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Question
Prove the following:
tan8θ − tan5θ − tan3θ = tan8θ · tan5θ · tan3θ
Solution
∵ 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
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