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प्रश्न
Prove that cos 5θ = 16 cos5θ – 20 cos3θ + 5 cos θ
उत्तर
cos5θ = cos(2θ + 3θ)
= cos 2θ cos 3θ – sin 2θ sin 3θ
= (2 cos2θ – 1)(4 cos3θ – 3 cos θ) – 2 sin θ cos θ(3 sin θ – 4 sin3θ)
= 8cos5θ – 6 cos3θ – 4 cos3θ + 3 cos θ – 6 sin2θ cos θ + 8 cos θ sin4θ
= 8 cos5θ – 6 cos3θ – 4 cos3θ + 3 cos θ – 6(1 – cos2θ) cos θ + 8 cos θ(1 – cos2θ)2
= 8 cos5θ – 6 cos3θ – 4 cos3θ + 3 cos θ – 6 cos θ + 6 cos3θ + 8 cos 0(1+ cos4θ – 2 cos2θ)
= 8 cos5θ – 6 cos3θ – 4 cos3θ + 3 cos θ – 6 cos θ + 6 cos3θ + 8 cos θ + 8 cos5θ – 16 cos3θ
= 16 cos5θ – 20 cos3θ + 5 cos θ
= R.H.S
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