if x=cos2th use the t formulae to show: sr((1+x)/(1-x))=|cot th|

We are tasked with proving that:

\[
\sqrt{\frac{1+x}{1-x}} = |\cot \theta|
\]

where \( x = \cos 2\theta \). Let's work through this step by step using trigonometric identities.

### Step 1: Use the double angle identity for cosine
We know the double angle identity for cosine is:

\[
\cos 2\theta = 2\cos^2\theta - 1
\]

Thus, \( x = \cos 2\theta \) can be written as:

\[
x = 2\cos^2\theta - 1
\]

### Step 2: Express \( \cos^2\theta \) in terms of \( x \)
Solving for \( \cos^2\theta \) from the above equation:

\[
x + 1 = 2\cos^2\theta
\]

\[
\cos^2\theta = \frac{x + 1}{2}
\]

### Step 3: Use the identity \( \sin^2\theta + \cos^2\theta = 1 \)
Now, using the Pythagorean identity \( \sin^2\theta + \cos^2\theta = 1 \), we can express \( \sin^2\theta \) in terms of \( \cos^2\theta \):

\[
\sin^2\theta = 1 - \cos^2\theta = 1 - \frac{x+1}{2} = \frac{1-x}{2}
\]

### Step 4: Express \( \cot \theta \) in terms of \( x \)
Recall that \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Using the expressions for \( \cos^2\theta \) and \( \sin^2\theta \), we have:

\[
\cot^2\theta = \frac{\cos^2\theta}{\sin^2\theta} = \frac{\frac{x+1}{2}}{\frac{1-x}{2}} = \frac{x+1}{1-x}
\]

Taking the square root on both sides:

\[
|\cot \theta| = \sqrt{\frac{x+1}{1-x}}
\]

### Step 5: Compare both sides
From the previous steps, we have:

\[
|\cot \theta| = \sqrt{\frac{x+1}{1-x}}
\]

This matches the required expression:

\[
\sqrt{\frac{1+x}{1-x}} = |\cot \theta|
\]

Thus, the proof is complete.

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

Here are 5 related questions:
1. How is the identity for \( \cos 2\theta \) derived?
2. Can we use a similar approach to prove an identity involving \( \tan \theta \)?
3. What other identities are related to \( \cot \theta \)?
4. How does the expression for \( \sin 2\theta \) differ from \( \cos 2\theta \)?
5. Can this proof be extended to other trigonometric functions?

**Tip:** Remember that understanding and using double-angle identities is key to simplifying many trigonometric problems.

This detailed solution shows how to prove the identity sqrt((1+x)/(1-x)) = |cot θ|, using the double angle formula for cos 2θ and trigonometric identities. Ideal for high school students studying trigonometry, this problem involves understanding the Pythagorean identity and properties of cotangent.

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