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प्रश्न
If cosα + cosβ = 0 = sinα + sinβ, then prove that cos2α + cos2β = -2cos(α + β).
[Hint: (cosα + cosβ)2 - (sinα + sinβ)2 = 0]
उत्तर
Given that: cosα + cosβ = 0
And sinα + sinβ = 0
∴ (cosα + cosβ)2 - (sinα + sinβ)2 = 0
⇒ (cos2α + cos2β + 2cosα cosβ) - (sin2α + sin2β + 2sinα sinβ) = 0
⇒ cos2α + cos2β + 2 cosα cosβ - sin2α - sin2β - 2 sinα sinβ = 0
⇒ (cos2α - sin2α) + (cos2β - sin2β) + 2(cosα cosβ - sinα sinβ) = 0
⇒ cos2α + cos2β + 2cos(α + β) = 0
Hence, cos2α + cos2β = -2cos(α + β).
Hence proved
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