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If sin α sin β - cos α cos β + 1 = 0, prove that 1 + cot α tan β = 0.
Given sin α sin β – cos α cos β + 1 = 0
⇒ -(cos α cos β – sin α sin β) + 1 = 0
We know that cos(A +B) = cosA cosB - sinA sinB
⇒ -cos(α + β) + 1 = 0
⇒ cos(α + β) = 1
We know that sin θ = √(1 – cos2 θ)
∴ sin(α + β) = 0 …(1)
Consider 1 + cot α tan β,
We know that sin(A ±B) = sinA cosB ± cosA sinB
= 0 = RHS
Hence, proved.
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