Question

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

Hint:

Another neat method is to square both sides of the given equation and use the identity \(\sin^2 A + \cos^2 A = 1\)!
Both methods are mathematically robust and yield full marks.

Answer 1

Step 1: Understanding the Question:
We are given a trigonometric relation: \(\cos A + \sin A = \sqrt{2} \cos A\). We need to prove that \(\cos A - \sin A = \sqrt{2} \sin A\).

Step 2: Key Formula or Approach:
We can rearrange the given equation to find a direct relation for \(\sin A\), and then substitute or manipulate algebraically to find \(\cos A - \sin A\).

Step 3: Detailed Explanation:

• Start with the given equation:
\[ \cos A + \sin A = \sqrt{2} \cos A \]

• Rearrange terms to group the cosines together:
\[ \sin A = \sqrt{2} \cos A - \cos A \]
\[ \sin A = (\sqrt{2} - 1) \cos A \]

• Rationalize to find \(\cos A\) in terms of \(\sin A\):
Multiply both sides by \((\sqrt{2} + 1)\):
\[ \sin A (\sqrt{2} + 1) = (\sqrt{2} - 1)(\sqrt{2} + 1) \cos A \]
\[ \sqrt{2} \sin A + \sin A = (2 - 1) \cos A \]
\[ \sqrt{2} \sin A + \sin A = \cos A \]

• Rearrange this new equation to isolate \(\cos A - \sin A\):
\[ \cos A - \sin A = \sqrt{2} \sin A \]


Step 4: Final Answer:
Hence, \(\cos A - \sin A = \sqrt{2} \sin A\) is successfully proved.