If sin α sin β - cos α cos β + 1 = 0, prove that 1 + cot α tan β = 0.

Question

Philoid · Accepted Answer

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.