How do you simplify sine squared theta divided by the quantity 1 minus sine squared theta?

To simplify the expression sin²(θ) / (1 – sin²(θ)), we start by recognizing a fundamental trigonometric identity. We know that:

1 – sin²(θ) = cos²(θ)

Using this identity, we can substitute cos²(θ) for 1 – sin²(θ) in the original expression:

sin²(θ) / (1 – sin²(θ)) = sin²(θ) / cos²(θ)

This expression can be rewritten using the definition of the tangent function, which states that:

tan(θ) = sin(θ) / cos(θ)

Therefore, we can further simplify:

sin²(θ) / cos²(θ) = (sin(θ) / cos(θ))² = tan²(θ)

So, the simplified form of the given expression is:

tan²(θ)