(sin alpha-cos alpha +1)÷(sin alpha + cos alpha -1)=(sin alpha + 1)÷(cos alpha)

Question

Prove that this eq is equal.

Answer 1

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.