Question 1 Report
Express \(\frac{1}{1 - \sin 45°}\) in surd form.
Answer Details
We can start by using the identity \(\sin 45° = \frac{\sqrt{2}}{2}\): \begin{align*} \frac{1}{1 - \sin 45°} &= \frac{1}{1 - \frac{\sqrt{2}}{2}} \\ &= \frac{1}{\frac{2 - \sqrt{2}}{2}} \\ &= \frac{2}{2 - \sqrt{2}} \cdot \frac{2 + \sqrt{2}}{2 + \sqrt{2}} \quad \text{(rationalizing the denominator)} \\ &= \frac{2(2 + \sqrt{2})}{2^2 - (\sqrt{2})^2} \\ &= \frac{2(2 + \sqrt{2})}{2} \\ &= \boxed{2 + \sqrt{2}} \end{align*} Therefore, the answer is option A: \(2 + \sqrt{2}\).