To prove that [tex]\(\cos \theta - \sin \theta = \sqrt{2} \sin \theta\)[/tex] given that [tex]\(\sin \theta + \cos \theta = \sqrt{2} \cos \theta\)[/tex], we can follow these steps:
1. Start from the given equation:
[tex]\[
\sin \theta + \cos \theta = \sqrt{2} \cos \theta
\][/tex]
2. Isolate [tex]\(\sin \theta\)[/tex] by subtracting [tex]\(\cos \theta\)[/tex] from both sides:
[tex]\[
\sin \theta = \sqrt{2} \cos \theta - \cos \theta
\][/tex]
3. Factor out [tex]\(\cos \theta\)[/tex]:
[tex]\[
\sin \theta = (\sqrt{2} - 1) \cos \theta
\][/tex]
4. Now, let's consider the equation we need to prove:
[tex]\[
\cos \theta - \sin \theta = \sqrt{2} \sin \theta
\][/tex]
5. Substitute the expression we found for [tex]\(\sin \theta\)[/tex] into this equation:
[tex]\[
\cos \theta - (\sqrt{2} - 1) \cos \theta = \sqrt{2} ((\sqrt{2} - 1) \cos \theta)
\][/tex]
6. Simplify the left-hand side by distributing and combining like terms:
[tex]\[
\cos \theta - \sqrt{2} \cos \theta + \cos \theta = \cos \theta (1 + 1 - \sqrt{2})
\][/tex]
[tex]\[
2 \cos \theta - \sqrt{2} \cos \theta
\][/tex]
7. Simplify the right-hand side by distributing [tex]\(\sqrt{2}\)[/tex]:
[tex]\[
\sqrt{2} (\sqrt{2} - 1) \cos \theta = (\sqrt{2} \cdot \sqrt{2} - \sqrt{2}) \cos \theta
\][/tex]
[tex]\[
(2 - \sqrt{2}) \cos \theta
\][/tex]
8. We see that both sides of the equation are equal:
[tex]\[
2 \cos \theta - \sqrt{2} \cos \theta = (2 - \sqrt{2}) \cos \theta
\][/tex]
Thus, we have proven that:
[tex]\[
\cos \theta - \sin \theta = \sqrt{2} \sin \theta
\][/tex]
Therefore, the statement is true based on the given condition.
