Finding a common denominator

Combine the two fractions by finding the common denominator, which is $\sin \theta (1-\cos \theta)$.

$\frac{\sin^2 \theta + (1-\cos \theta)^2}{\sin \theta (1-\cos \theta)}$

Expanding and simplifying

Expand the numerator using $(1-\cos \theta)^2 = 1 - 2\cos \theta + \cos^2 \theta$.

$\frac{\sin^2 \theta + 1 - 2\cos \theta + \cos^2 \theta}{\sin \theta (1-\cos \theta)}$

Applying trigonometric identities

Use the identity $\sin^2 \theta + \cos^2 \theta = 1$ to simplify the numerator.

$\frac{1 + 1 - 2\cos \theta}{\sin \theta (1-\cos \theta)} = \frac{2 - 2\cos \theta}{\sin \theta (1-\cos \theta)}$

Final reduction

Factor out 2 from the numerator and cancel the common term $(1-\cos \theta)$.

$\frac{2(1-\cos \theta)}{\sin \theta (1-\cos \theta)} = \frac{2}{\sin \theta} = 2\csc \theta$