Question: Prove the identity (sin alpha-cos alpha +1)÷(sin alpha + cos alpha -1)=(sin alpha + 1)÷(cos alpha).
Cross-multiply, to get
(sin(α) – cos(α) + 1)*cos(α) = (sin(α) + cos(α) – 1)*(sin(α) + 1)
Multiplying out and re-arranging,
sin(α).cos(α) – cos^2(α) + cos(α) = sin*2(α) + sin(α).cos(α) – sin(α) + sin(α) + cos(α) – 1
sin(α).cos(α) – cos^2(α) + cos(α) = sin*2(α) – 1 + sin(α).cos(α) + cos(α)
sin(α).cos(α) – cos^2(α) + cos(α) = -cos*2(α) + sin(α).cos(α) + cos(α)
Since lhs = rhs now, then the identity is proved.