Express \(\frac{1}{1 - \sin 45°}\) in surd form.
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: \(2 + \sqrt{2}\).
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