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The problem requires proving the trigonometric identity:

\[
\frac{\sin x + 1}{\cos x + \cot x} = \tan x
\]

### Proof:

1. **Rewrite cotangent in terms of sine and cosine**:
   \[
   \cot x = \frac{\cos x}{\sin x}
   \]
   Substituting this into the denominator of the left-hand side:

   \[
   \frac{\sin x + 1}{\cos x + \frac{\cos x}{\sin x}} = \frac{\sin x + 1}{\frac{\cos x \sin x + \cos x}{\sin x}} = \frac{\sin x + 1}{\frac{\cos x(1 + \sin x)}{\sin x}}
   \]

2. **Simplify the fraction**:
   Cancel out \((1 + \sin x)\) from the numerator and denominator:
   \[
   = \frac{\sin x}{\cos x} = \tan x
   \]

Thus, the left-hand side simplifies to \(\tan x\), proving the identity is true.

Would you like more details or have any questions?

Here are 5 related questions:
1. How can cotangent be expressed in terms of sine and cosine?
2. What is the identity for tangent in terms of sine and cosine?
3. How can fractions with trigonometric functions be simplified?
4. What other identities involve tangent and cotangent?
5. How can verifying identities help in solving complex trigonometric equations?

**Tip:** Always start by expressing all trigonometric functions in terms of sine and cosine when proving identities!

Prove that the following identity is true: (sin x + 1) / (cos x + cot x) = tan x

Learn how to prove the trigonometric identity (sin x + 1) / (cos x + cot x) = tan x with a step-by-step solution. This solution involves rewriting cotangent, simplifying fractions, and using fundamental trigonometric identities. Ideal for students in Grades 11-12 studying advanced trigonometry.

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