Alternative Direct Substitution Approach

1. Start from the given equation:
   \[ \cos\theta + \sin\theta = \sqrt{2}\cos\theta. \]
2. Rearranging, we get:
   \[ \sin\theta = (\sqrt{2}-1)\cos\theta. \]
3. Now consider what we need to prove:
   \[ \cos\theta – \sin\theta = \sqrt{2}\sin\theta. \]
4. Substitute \(\sin\theta = (\sqrt{2}-1)\cos\theta\) into the left-hand side:
   \[ \cos\theta – \sin\theta = \cos\theta – (\sqrt{2}-1)\cos\theta = \cos\theta(2-\sqrt{2}). \]
5. Substitute \(\sin\theta = (\sqrt{2}-1)\cos\theta\) into the right-hand side:
   \[ \sqrt{2}\sin\theta = \sqrt{2}(\sqrt{2}-1)\cos\theta = (2-\sqrt{2})\cos\theta. \]
6. Hence, both sides are equal:
   \[ \cos\theta – \sin\theta = (2-\sqrt{2})\cos\theta \quad\text{and}\quad \sqrt{2}\sin\theta = (2-\sqrt{2})\cos\theta. \]
   Since the left-hand side and the right-hand side simplify to the same expression, the proof is complete.