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: Understanding the Question:
We are given the relation \(\cos \theta + \sin \theta = \sqrt{2} \cos \theta\). We need to prove that \(\cos \theta - \sin \theta = \sqrt{2} \sin \theta\).

Step 2: Key Formula or Approach:
We can solve this problem using algebraic manipulation and squaring. A very elegant method is to use the algebraic identity:
\[ (a + b)^2 + (a - b)^2 = 2(a^2 + b^2) \]
By setting \(a = \cos \theta\) and \(b = \sin \theta\), we can substitute the known expression and solve for the unknown expression.

Step 3: Detailed Explanation:
1. Let \(a = \cos \theta\) and \(b = \sin \theta\).
2. Consider the sum of the squares of the two expressions:
\[ X = (\cos \theta + \sin \theta)^2 + (\cos \theta - \sin \theta)^2 \]
3. Expand both squared binomial terms:
\[ X = (\cos^2 \theta + \sin^2 \theta + 2\sin \theta\cos \theta) + (\cos^2 \theta + \sin^2 \theta - 2\sin \theta\cos \theta) \]
4. Combine like terms:
The cross terms \(2\sin \theta\cos \theta\) and \(-2\sin \theta\cos \theta\) cancel out:
\[ X = 2(\cos^2 \theta + \sin^2 \theta) \]
Since we know \(\cos^2 \theta + \sin^2 \theta = 1\):
\[ (\cos \theta + \sin \theta)^2 + (\cos \theta - \sin \theta)^2 = 2 \quad \text{--- (Equation 1)} \]
5. Substitute the given value \(\cos \theta + \sin \theta = \sqrt{2} \cos \theta\) into Equation 1:
\[ (\sqrt{2} \cos \theta)^2 + (\cos \theta - \sin \theta)^2 = 2 \]
\[ 2 \cos^2 \theta + (\cos \theta - \sin \theta)^2 = 2 \]
6. Isolate the target squared term:
\[ (\cos \theta - \sin \theta)^2 = 2 - 2 \cos^2 \theta \]
Factor out 2 on the right-hand side:
\[ (\cos \theta - \sin \theta)^2 = 2(1 - \cos^2 \theta) \]
7. Use the identity \(1 - \cos^2 \theta = \sin^2 \theta\):
\[ (\cos \theta - \sin \theta)^2 = 2 \sin^2 \theta \]
8. Take the square root on both sides:
\[ \cos \theta - \sin \theta = \sqrt{2} \sin \theta \]
This completes the proof.

Step 4: Final Answer:
The given equation has been used to prove the target equation. Thus, the proof is completed.