https://file.aiquickdraw.com/mathbot/file/1730383225074h79yjhoa.jpg

The uploaded image shows three trigonometric expressions, likely intended for proof or simplification.

Here’s a breakdown of each:

### Expression (i):
\[
\tan^{-1} \left( \frac{\sin A \cos B}{\cos A + \sin B} \right)
\]

### Expression (ii):
\[
\left[ \tan^{-1} \left( \frac{\pi}{4 - 2} \right) \right] = \text{to prove}
\]

### Expression (iii):
\[
2 \tan^{-1} \left( \frac{A}{\tan \frac{B}{2} \tan \frac{A}{2}} \right) = \text{to prove}
\]

Please confirm if this is accurate, or if you’d like me to start solving or proving each one individually. 

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Would you like to go through detailed steps? Here are five related questions:

1. What identities would simplify the given expressions in trigonometry proofs?
2. How do inverse trigonometric functions behave with addition and multiplication?
3. How do you transform a tangent function in terms of sine and cosine?
4. What are common strategies for proving expressions involving \(\tan^{-1}\)?
5. How would complex trigonometric proofs simplify with angle-sum or difference formulas?

**Tip:** For inverse trigonometric proofs, rewriting terms in terms of sine and cosine often helps clarify the solution path.

Prove the trigonometric expressions as shown in the image.

Explore the proof of trigonometric expressions involving inverse functions and addition/subtraction formulas. This solution explains how to approach complex trigonometric proofs using inverse trigonometric identities. Suitable for students in Grades 11-12.

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