Step 1: Identify the structure of the expression.
We need to simplify $\frac{1}{1+\sin\theta} + \frac{1}{1-\sin\theta}$. The two fractions share a pair of conjugate denominators, which suggests combining them under a common denominator.
Step 2: Find the common denominator.
The LCD is $(1+\sin\theta)(1-\sin\theta)$. Rewriting: \[ \frac{1-\sin\theta + 1+\sin\theta}{(1+\sin\theta)(1-\sin\theta)} = \frac{2}{(1+\sin\theta)(1-\sin\theta)}. \]
Step 3: Simplify the numerator.
The numerator $(1-\sin\theta)+(1+\sin\theta) = 2$. The $\sin\theta$ terms cancel.
Step 4: Simplify the denominator using a difference-of-squares identity.
$(1+\sin\theta)(1-\sin\theta) = 1 - \sin^2\theta$. This is a standard algebraic identity $(a+b)(a-b)=a^2-b^2$ with $a=1, b=\sin\theta$.
Step 5: Apply the Pythagorean identity.
By the fundamental identity $\sin^2\theta + \cos^2\theta = 1$, we get $1 - \sin^2\theta = \cos^2\theta$. So the expression becomes $\frac{2}{\cos^2\theta}$.
Step 6: Express in terms of secant and state the answer.
Since $\sec\theta = \frac{1}{\cos\theta}$, we have $\frac{1}{\cos^2\theta} = \sec^2\theta$. Therefore: \[ \frac{1}{1+\sin\theta}+\frac{1}{1-\sin\theta} = 2\sec^2\theta. \] \[ \boxed{2\sec^2\theta} \]