Question

If \(\cos \theta + \sin \theta = \sqrt{2} \cos \theta\), then prove that \(\cos \theta - \sin \theta = \sqrt{2} \sin \theta\).

• A. Proof Completed
• B. Identity is Incorrect
• C. Incomplete Algebraic Steps
• D. Cannot be Determined

Hint:

An alternative way to solve this is to rearrange the original equation first:
\[ \sin \theta = \sqrt{2}\cos\theta - \cos\theta = (\sqrt{2} - 1)\cos\theta \]
Multiply both sides by \((\sqrt{2} + 1)\):
\[ (\sqrt{2} + 1)\sin\theta = (2 - 1)\cos\theta = \cos\theta \]
\[ \sqrt{2}\sin\theta + \sin\theta = \cos\theta \]
\[ \cos\theta - \sin\theta = \sqrt{2}\sin\theta \]
This algebraic method is also extremely fast and intuitive!

Answer 1

The Correct Option is A

Step 1: Understand what is being asked.
We are given $\cos\theta+\sin\theta=\sqrt{2}\cos\theta$ and must prove $\cos\theta-\sin\theta=\sqrt{2}\sin\theta$. Instead of squaring both sides, let us rearrange the given equation to isolate $\sin\theta$, and build up to the required expression algebraically.

Step 2: Rearrange the given equation.
\[ \cos\theta+\sin\theta = \sqrt{2}\cos\theta \]
Move $\cos\theta$ to the right:
\[ \sin\theta = \sqrt{2}\cos\theta - \cos\theta = (\sqrt{2}-1)\cos\theta \]
Step 3: Multiply both sides by $(\sqrt{2}+1)$ to simplify the surd.
We choose this multiplier because $(\sqrt{2}-1)(\sqrt{2}+1)=2-1=1$, a clean whole number:
\[ (\sqrt{2}+1)\sin\theta = (\sqrt{2}-1)(\sqrt{2}+1)\cos\theta = (2-1)\cos\theta = \cos\theta \]
Step 4: Expand the left side and rearrange to match the required form.
\[ \cos\theta = (\sqrt{2}+1)\sin\theta = \sqrt{2}\sin\theta+\sin\theta \]
Move $\sin\theta$ to the left side:
\[ \cos\theta - \sin\theta = \sqrt{2}\sin\theta \]
Final Answer:
This is exactly the relation we were asked to prove, so the proof is complete.
\[ \boxed{\cos\theta-\sin\theta=\sqrt{2}\sin\theta} \]