Given Equation:
The given equation is:
1 + 2sinθ + (sin^2θ / cos^2θ)

To prove:
We need to prove that:
(secθ * tanθ) / (secθ - tanθ)

Solution:

Step 1: Simplify the given equation
1 + 2sinθ + (sin^2θ / cos^2θ)
= (cos^2θ + 2sinθcos^2θ + sin^2θ) / cos^2θ
= (cos^2θ + 2sinθcos^2θ + sin^2θ) / cos^2θ
= [(cos^2θ + sin^2θ) + 2sinθcos^2θ] / cos^2θ
= [1 + sinθ(2cos^2θ)] / cos^2θ
= (1 + 2sinθcos^2θ) / cos^2θ

Step 2: Rewrite sinθcos^2θ as sinθcosθ * cosθ
= (1 + 2sinθcosθ * cosθ) / cos^2θ
= (1 + sinθ * 2cosθ) / cos^2θ

Step 3: Rewrite 2cosθ as 2sinθ / sinθcosθ
= (1 + sinθ * 2sinθ / sinθcosθ) / cos^2θ
= (1 + 2sin^2θ / sinθcosθ) / cos^2θ

Step 4: Rewrite sinθcosθ as 1/2 * 2sinθcosθ
= (1 + 2sin^2θ / 1/2 * 2sinθcosθ) / cos^2θ
= (1 + 4sin^2θ / 2sinθcosθ) / cos^2θ

Step 5: Rewrite 2sinθcosθ as sin2θ
= (1 + 4sin^2θ / sin2θ) / cos^2θ
= (1 + 4sin^2θ / 2sinθcosθ) / cos^2θ

Step 6: Simplify the expression
= (1 + 2sinθ / cosθ) / cos^2θ
= (1 + 2sinθ / cosθ) / (cosθ / cos^2θ)
= (1 + 2sinθ / cosθ) * (cos^2θ / cosθ)
= (1 + 2sinθ) * cosθ / cosθ
= 1 + 2sinθ

Step 7: Simplify the expression secθ - tanθ
= 1/cosθ - sinθ / cosθ
= (1 - sinθ) / cosθ

Step 8: Simplify the expression (secθ * tanθ) / (secθ - tanθ)
= [(1/cosθ) * (sinθ / cosθ)] / [(1 - sinθ) / cosθ]
= (sinθ / cos^2θ) / ((1 - sinθ