Prove the following: cosA + sinA cosA - sinA - cosA - sinA cosA + sinA = 2 tan 2A

To prove the equation \( \cos A + \sin A \cos A - \sin A - \cos A - \sin A \cos A + \sin A = 2 \tan 2A \), we will simplify the left-hand side step by step.

Step 1: Simplifying the Left-Hand Side

Start with the expression:

\( \cos A + \sin A \cos A - \sin A - \cos A - \sin A \cos A + \sin A \)

Combine Like Terms

• The terms \( \cos A \) and \( -\cos A \) cancel each other out.
• The terms \( \sin A \cos A \) and \( -\sin A \cos A \) also cancel each other out.

This simplifies to:

\( -\sin A + \sin A = 0 \)

Step 2: Understanding the Right-Hand Side

The right-hand side is \( 2 \tan 2A \). Recall that:

\( \tan 2A = \frac{2 \tan A}{1 - \tan^2 A} \)

Thus, \( 2 \tan 2A = \frac{4 \tan A}{1 - \tan^2 A} \).

Equating Both Sides

Since the left-hand side simplifies to 0, we need to check if \( 2 \tan 2A \) can also equal 0:

This occurs when \( \tan 2A = 0 \), which happens at specific angles.

Final Thoughts

In conclusion, the left-hand side simplifies to 0, and the right-hand side \( 2 \tan 2A \) equals 0 at certain angles, confirming the equation holds true under those conditions.