sin(alpha + beta) / sin(alpha) cos(beta) = 1 + cot(alpha) tan(beta), Establish the identity

Question

if you use a certain formula to get an answer please show which formula you use so I know how you got a specific answer, thank you!

Answer 1

We can use the sum formula for sine as follows:

sin(A + B) / (sinAcosB) = 1 + cotAtanB

(sinAcosB + sinBcosA) / (sinAcosB) = 1 + cotAtanB (sum formula for sine)

1 + sinBcosA / sinAcosB = 1 + cotAtanB (dividing both terms in numerator on left by denominator)

1 + (cosA)/(sinA) · (sinB)/(cosA) = 1 + cotAtanB (properties of fractions / commutativity of multiplication)

The left = right by quotient identities for tangent and cotangent, verifying the given identity.