if cosA-sinA =  **√2**sinA, then show sinA+cosA = **√2**cosA

We are given the equation:

\[
\cos A - \sin A = \sqrt{2} \sin A
\]

We need to show that:

\[
\sin A + \cos A = \sqrt{2} \cos A
\]

### Step 1: Manipulate the given equation
From the given equation, we can isolate \(\cos A\) on one side:

\[
\cos A = \sqrt{2} \sin A + \sin A
\]

Factor out \(\sin A\) on the right-hand side:

\[
\cos A = \sin A (\sqrt{2} + 1)
\]

### Step 2: Use this expression for \(\cos A\)
Now that we have \(\cos A = \sin A (\sqrt{2} + 1)\), substitute this expression into the equation we are trying to prove:

\[
\sin A + \cos A = \sqrt{2} \cos A
\]

Substitute the value of \(\cos A\):

\[
\sin A + \sin A (\sqrt{2} + 1) = \sqrt{2} [\sin A (\sqrt{2} + 1)]
\]

### Step 3: Simplify the left-hand side
On the left-hand side, factor out \(\sin A\):

\[
\sin A \left[ 1 + (\sqrt{2} + 1) \right] = \sqrt{2} \sin A (\sqrt{2} + 1)
\]

Simplify inside the brackets:

\[
\sin A \left[ \sqrt{2} + 2 \right] = \sqrt{2} \sin A (\sqrt{2} + 1)
\]

### Step 4: Verify both sides
Both sides are now equivalent, as the expression \(\sqrt{2} + 2\) on the left-hand side simplifies to match the right-hand side. Thus, we have shown:

\[
\sin A + \cos A = \sqrt{2} \cos A
\]

### Conclusion
The identity has been proven.

Would you like more details on any step, or do you have any questions?

Here are 5 related questions to expand on this:

1. Can you find the value of \(\sin A\) and \(\cos A\) if \(\cos A - \sin A = \sqrt{2} \sin A\)?
2. How does factoring help in simplifying trigonometric expressions?
3. How can you prove similar trigonometric identities using known formulas?
4. What is the significance of identities in solving trigonometric equations?
5. How would you solve for \(A\) if both sides of the equation contained different trigonometric functions?

**Tip:** Always try to factor or isolate variables when dealing with trigonometric identities.

If cosA - sinA = √2 sinA, then show sinA + cosA = √2 cosA.

Learn how to prove the trigonometric identity starting with cosA - sinA = √2 sinA and showing that it leads to sinA + cosA = √2 cosA. This step-by-step solution uses factoring and trigonometric identities, ideal for high school students in Grades 10-12.

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