Question

If $\sin \theta + \cos \theta = \sqrt{2} \cos \theta$, then $\cos \theta - \sin \theta =$

• A. $\sqrt{2} \sin \theta$
• B. $-\sqrt{2} \sin \theta$
• C. $\sqrt{2} \cos \theta$
• D. $-\sqrt{2} \cos \theta$

Hint:

There is a famous identity: if $a \cos\theta + b \sin\theta = c$, then $b \cos\theta - a \sin\theta = \pm \sqrt{a^2+b^2-c^2}$. Here, $1^2 + 1^2 - (\sqrt{2}\cos\theta)^2 = 2 - 2\cos^2\theta = 2\sin^2\theta$, giving $\sqrt{2}\sin\theta$.

Answer 1

The Correct Option is A

Step 1: Concept
We use trigonometric identity transformations or algebraic manipulations on the equation $\sin \theta + \cos \theta = \sqrt{2} \cos \theta$.

Step 2: Meaning
We want to express $\cos \theta - \sin \theta$ in terms of $\sin \theta$ or $\cos \theta$.

Step 3: Analysis
From the given equation: \[ \sin \theta = \sqrt{2} \cos \theta - \cos \theta \implies \sin \theta = (\sqrt{2} - 1) \cos \theta \] Multiply both sides by $(\sqrt{2} + 1)$: \[ (\sqrt{2} + 1) \sin \theta = (\sqrt{2} + 1)(\sqrt{2} - 1) \cos \theta \] \[ \sqrt{2} \sin \theta + \sin \theta = (2 - 1) \cos \theta = \cos \theta \] Rearranging the terms: \[ \cos \theta - \sin \theta = \sqrt{2} \sin \theta \]

Step 4: Conclusion
Thus, the expression $\cos \theta - \sin \theta$ is equal to $\sqrt{2} \sin \theta$. Final Answer: (A)