We are asked to prove the trigonometric identity:

\[
\frac{\csc x}{\cot x + \tan x} = \cos x
\]

### Step-by-step proof:

1. **Express in terms of basic trigonometric functions:**

   Recall the definitions of cosecant, cotangent, and tangent in terms of sine and cosine:
   \[
   \csc x = \frac{1}{\sin x}, \quad \cot x = \frac{\cos x}{\sin x}, \quad \tan x = \frac{\sin x}{\cos x}
   \]
   
   Substitute these into the left-hand side (LHS):
   \[
   \frac{\csc x}{\cot x + \tan x} = \frac{\frac{1}{\sin x}}{\frac{\cos x}{\sin x} + \frac{\sin x}{\cos x}}
   \]

2. **Simplify the denominator:**

   To simplify \(\frac{\cos x}{\sin x} + \frac{\sin x}{\cos x}\), get a common denominator:
   \[
   \frac{\cos x}{\sin x} + \frac{\sin x}{\cos x} = \frac{\cos^2 x + \sin^2 x}{\sin x \cos x}
   \]
   
   Use the Pythagorean identity \(\cos^2 x + \sin^2 x = 1\):
   \[
   \frac{1}{\sin x \cos x}
   \]

3. **Rewrite the expression:**

   Now, substitute this simplified denominator back into the expression:
   \[
   \frac{\frac{1}{\sin x}}{\frac{1}{\sin x \cos x}} = \frac{1}{\sin x} \cdot \frac{\sin x \cos x}{1} = \cos x
   \]

Thus, the left-hand side simplifies to \(\cos x\), which is exactly the right-hand side (RHS).

### Conclusion:
\[
\boxed{\frac{\csc x}{\cot x + \tan x} = \cos x}
\]

This proves the given identity.

---

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

Here are 5 related questions to expand on this:

1. How do we derive the Pythagorean identity \(\cos^2 x + \sin^2 x = 1\)?
2. What is the relationship between sine, cosine, and the unit circle?
3. How can we prove \(\csc x = \frac{1}{\sin x}\) using the unit circle?
4. How can we solve trigonometric equations involving both sine and cosine functions?
5. What are the conditions under which trigonometric identities hold?

**Tip:** When dealing with trigonometric identities, expressing all functions in terms of sine and cosine often simplifies the problem.

Learn how to prove the trigonometric identity cosec(x)/(cot(x)+tan(x)) = cos(x) through a step-by-step approach. This problem involves fundamental trigonometric identities and the Pythagorean theorem. Suitable for students in Grades 10-12.

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