$$ \tan^{-1} $$$$ \frac{\left(\cos x\right)}{\left(1-\sin x\right)} $$

You provided the following expression:

\[
\tan^{-1} \left( \frac{\cos x}{1 - \sin x} \right)
\]

This expression can be simplified using a trigonometric identity. Let's work through the simplification.

### Step-by-step Simplification:

1. **Recognize the tangent identity**:
   \[
   \tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos\theta}{\sin\theta}
   \]

2. **Substitute the complementary angle identity** for the tangent:
   Using $ \tan^{-1}(\cdot) $, we can rewrite the expression inside the inverse tangent as:
   \[
   \frac{\cos x}{1 - \sin x}
   \]
   This is equivalent to $ \tan\left(\frac{\pi}{2} - x\right) $, based on the known trigonometric identity:
   \[
   \frac{\cos x}{1 - \sin x} = \tan\left(\frac{x}{2}\right)
   \]

3. **Final result**:
   Hence, the original expression simplifies to:
   \[
   \tan^{-1}\left( \frac{\cos x}{1 - \sin x} \right) = \frac{x}{2}
   \]

So, the simplified form of the given expression is:

\[
\boxed{\frac{x}{2}}
\]

---

Would you like more details on any step? Let me know if you have questions!

Here are 5 related questions for further exploration:
1. How is the arctangent function related to inverse trigonometric functions in general?
2. What are other trigonometric identities that involve arctan?
3. Can you solve an equation involving $ \tan^{-1} $?
4. How does the simplification of trigonometric expressions benefit calculus?
5. What is the geometric interpretation of $ \tan^{-1} $?

**Tip**: Understanding the relationships between the different trigonometric functions can simplify complex expressions significantly!

$$ \tan^{-1} \frac{\left(\cos x\right)}{\left(1-\sin x\right)} $$

Learn how to simplify the inverse tangent expression $$ \tan^{-1} \frac{\cos x}{1 - \sin x} $$ using key trigonometric identities. This solution explores the use of complementary angle identities and inverse trigonometric functions. Suitable for high school students in Grades 10-12.

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