Let's solve the given equation step by step.
The equation provided is:
[tex]\[ \cot A = \pm \sqrt{\frac{1 + \cos 2x}{1 - \cos 2x}} \][/tex]
Here's a detailed breakdown of how we can understand and derive this expression:
### Step 1: Understanding Trigonometric Identities
First, let's recall some relevant trigonometric identities:
- The double angle formula for cosine: [tex]\(\cos 2x = \cos^2 x - \sin^2 x\)[/tex]
- The Pythagorean identity: [tex]\(\sin^2 x + \cos^2 x = 1\)[/tex]
However, we don't need to simplify these further as we are provided with the expression directly.
### Step 2: Analyzing the Expression
Given the expression involves [tex]\(\cos (2x)\)[/tex], observe that it uses the double angle formula in a more generalized form. We need to reinterpret how [tex]\(\cot A\)[/tex] can be expressed using [tex]\(\cos 2x\)[/tex].
### Step 3: Simplifying the Expression
The expression under the square root is:
[tex]\[
\sqrt{\frac{1 + \cos 2x}{1 - \cos 2x}}
\][/tex]
### Step 4: Result Interpretation
This step involves understanding the obtained result:
[tex]\[
\cot A = \pm \sqrt{\frac{1 + \cos 2x}{1 - \cos 2x}}
\][/tex]
Note the plus-minus sign ([tex]\(\pm\)[/tex]). It indicates that [tex]\( \cot A \)[/tex] can take either the positive or negative value of the square root expression.
### Conclusion
The final expression [tex]\( \cot A \)[/tex] in terms of [tex]\( \cos 2x \)[/tex] is:
[tex]\[
\cot A = \pm \sqrt{\frac{1 + \cos 2x}{1 - \cos 2x}}
\][/tex]
Thus, we identified that [tex]\( \cot A \)[/tex] can be expressed using [tex]\( \cos 2x \)[/tex] by the transformation into the square root form. This expression is essential in trigonometric studies, especially while dealing with identities and simplifying equations involving [tex]\( \cot \)[/tex] and [tex]\( \cos \)[/tex].
