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The equation in the image asks to prove the following identity:

\[
\frac{\cos^2 \alpha + \cot \alpha}{\cos^2 \alpha - \cot \alpha} = \frac{\cos^2 \alpha \tan \alpha + 1}{\cos^2 \alpha \tan \alpha - 1}
\]

### Steps to Prove the Identity:
We will work on each side of the equation separately and attempt to transform one side into the other.

#### Left-Hand Side (LHS):
\[
\frac{\cos^2 \alpha + \cot \alpha}{\cos^2 \alpha - \cot \alpha}
\]
We can express \(\cot \alpha\) in terms of sine and cosine:
\[
\cot \alpha = \frac{\cos \alpha}{\sin \alpha}
\]
Substituting this into the expression, the LHS becomes:
\[
\frac{\cos^2 \alpha + \frac{\cos \alpha}{\sin \alpha}}{\cos^2 \alpha - \frac{\cos \alpha}{\sin \alpha}}
\]
To simplify the fractions, multiply both the numerator and the denominator by \(\sin \alpha\):
\[
\frac{\sin \alpha (\cos^2 \alpha + \frac{\cos \alpha}{\sin \alpha})}{\sin \alpha (\cos^2 \alpha - \frac{\cos \alpha}{\sin \alpha})} = \frac{\sin \alpha (\cos^2 \alpha \sin \alpha + \cos \alpha)}{\sin \alpha (\cos^2 \alpha \sin \alpha - \cos \alpha)}
\]
Simplifying:
\[
= \frac{\cos \alpha (\sin \alpha \cos \alpha + 1)}{\cos \alpha (\sin \alpha \cos \alpha - 1)}
\]
Canceling out \(\cos \alpha\) from both numerator and denominator:
\[
= \frac{\sin \alpha \cos \alpha + 1}{\sin \alpha \cos \alpha - 1}
\]

#### Right-Hand Side (RHS):
\[
\frac{\cos^2 \alpha \tan \alpha + 1}{\cos^2 \alpha \tan \alpha - 1}
\]
Recall that \(\tan \alpha = \frac{\sin \alpha}{\cos \alpha}\). Substituting this into the RHS:
\[
= \frac{\cos^2 \alpha \cdot \frac{\sin \alpha}{\cos \alpha} + 1}{\cos^2 \alpha \cdot \frac{\sin \alpha}{\cos \alpha} - 1}
\]
Simplifying:
\[
= \frac{\cos \alpha \sin \alpha + 1}{\cos \alpha \sin \alpha - 1}
\]

### Conclusion:
Both the LHS and the RHS simplify to the same expression:
\[
\frac{\sin \alpha \cos \alpha + 1}{\sin \alpha \cos \alpha - 1}
\]
Thus, the identity is proven.

Do you want further details or clarifications? Here are some related questions for practice:

1. What is the cotangent identity in terms of sine and cosine?
2. How can you simplify expressions involving both cotangent and tangent?
3. What is the general approach to proving trigonometric identities?
4. Can you prove the identity using a different method, like transforming the RHS instead?
5. How would the identity change if we used a different trigonometric function (like secant or cosecant)?

**Tip:** When proving trigonometric identities, try transforming one side step by step and avoid working on both sides simultaneously to maintain clarity.

Prove the trigonometric identity: (cos^2(α) + cot(α)) / (cos^2(α) - cot(α)) = (cos^2(α) * tan(α) + 1) / (cos^2(α) * tan(α) - 1)

Step-by-step solution to prove the trigonometric identity (cos^2(α) + cot(α)) / (cos^2(α) - cot(α)) = (cos^2(α) * tan(α) + 1) / (cos^2(α) * tan(α) - 1). This guide covers the use of cotangent and tangent formulas, as well as simplifying expressions. Suitable for students in Grades 10-12.

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