(cota-tana)/(cota+tana)=cos(2a)
a = alpha
prove the identity

We are tasked with proving the trigonometric identity:

\[
\frac{\cot(\alpha) - \tan(\alpha)}{\cot(\alpha) + \tan(\alpha)} = \cos(2\alpha)
\]

### Step 1: Express cotangent and tangent in terms of sine and cosine

First, recall that:
\[
\cot(\alpha) = \frac{\cos(\alpha)}{\sin(\alpha)}
\quad \text{and} \quad
\tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)}
\]

Now substitute these into the left-hand side of the equation:

\[
\frac{\frac{\cos(\alpha)}{\sin(\alpha)} - \frac{\sin(\alpha)}{\cos(\alpha)}}{\frac{\cos(\alpha)}{\sin(\alpha)} + \frac{\sin(\alpha)}{\cos(\alpha)}}
\]

### Step 2: Simplify the expression

We simplify both the numerator and the denominator by getting a common denominator within each.

#### Numerator:
\[
\frac{\frac{\cos^2(\alpha) - \sin^2(\alpha)}{\sin(\alpha)\cos(\alpha)}}
\]

#### Denominator:
\[
\frac{\frac{\cos^2(\alpha) + \sin^2(\alpha)}{\sin(\alpha)\cos(\alpha)}}
\]

### Step 3: Cancel common terms

Since both the numerator and denominator have the common factor \(\frac{1}{\sin(\alpha)\cos(\alpha)}\), these factors cancel out, and we are left with:

\[
\frac{\cos^2(\alpha) - \sin^2(\alpha)}{\cos^2(\alpha) + \sin^2(\alpha)}
\]

### Step 4: Use the Pythagorean identity

Recall the Pythagorean identity:

\[
\cos^2(\alpha) + \sin^2(\alpha) = 1
\]

Thus, the expression simplifies to:

\[
\frac{\cos^2(\alpha) - \sin^2(\alpha)}{1}
\]

This is simply:

\[
\cos(2\alpha)
\]

### Conclusion:

We have shown that the left-hand side of the original equation simplifies to \(\cos(2\alpha)\), so the identity is proven:

\[
\frac{\cot(\alpha) - \tan(\alpha)}{\cot(\alpha) + \tan(\alpha)} = \cos(2\alpha)
\]

Would you like more details or have any questions?

Here are five related questions to explore further:
1. How can you derive other double-angle identities like \(\sin(2\alpha)\) or \(\tan(2\alpha)\)?
2. What is the geometric interpretation of trigonometric identities?
3. Can the identity be proven using a different method, such as using Euler's formula?
4. How does this identity help in solving trigonometric equations?
5. How are identities involving \(\cot\) and \(\tan\) useful in calculus?

**Tip:** Mastering the double-angle identities like \(\cos(2\alpha)\) helps simplify many advanced trigonometric problems in calculus and physics.

Prove the identity (cot(alpha) - tan(alpha)) / (cot(alpha) + tan(alpha)) = cos(2alpha)

Learn how to prove the trigonometric identity (cot(alpha) - tan(alpha)) / (cot(alpha) + tan(alpha)) = cos(2alpha) step-by-step. This problem explores key trigonometric concepts such as cotangent, tangent, and double-angle formulas. Suitable for students in Grades 11-12.

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