To determine whether the given equation [tex]\(\tan^2(x) = \frac{1 + \cos(2x)}{1 - \cos(2x)}\)[/tex] is true, let's break down the left and right sides of the equation using trigonometric identities and see if they are equivalent.
### Step 1: Understanding the Right Side of the Equation
We start with the right side of the equation:
[tex]\[ \frac{1 + \cos(2x)}{1 - \cos(2x)} \][/tex]
### Step 2: Using Double-Angle Identities
We can use the double-angle identities for cosine to transform the terms:
[tex]\[ \cos(2x) = 2\cos^2(x) - 1 \][/tex]
### Step 3: Substitute Double-Angle Identity into the Equation
Substitute:
[tex]\[ 1 + \cos(2x) \][/tex]
[tex]\[ = 1 + (2\cos^2(x) - 1) \][/tex]
[tex]\[ = 2\cos^2(x) \][/tex]
Similarly:
[tex]\[ 1 - \cos(2x) \][/tex]
[tex]\[ = 1 - (2\cos^2(x) - 1) \][/tex]
[tex]\[ = 2 - 2\cos^2(x) \][/tex]
[tex]\[ = 2(1 - \cos^2(x)) \][/tex]
[tex]\[ = 2\sin^2(x) \][/tex]
### Step 4: Simplify the Right Side
Now, we substitute these back into the right side of the original equation:
[tex]\[ \frac{1 + \cos(2x)}{1 - \cos(2x)} = \frac{2\cos^2(x)}{2\sin^2(x)} \][/tex]
This simplifies to:
[tex]\[ \frac{\cos^2(x)}{\sin^2(x)} = \cot^2(x) \][/tex]
### Step 5: Compare with the Left Side of the Equation
The left side of the original equation is [tex]\(\tan^2(x)\)[/tex]. Recall the fundamental trigonometric identity involving tangent and cotangent:
[tex]\[ \tan(x) = \frac{1}{\cot(x)} \][/tex]
Which gives us:
[tex]\[ \tan^2(x) = \cot^2(x) \][/tex]
### Conclusion
The equation [tex]\(\tan^2(x) = \frac{1 + \cos(2x)}{1 - \cos(2x)}\)[/tex] holds true according to the identities and simplifications done above.
Thus, the correct answer is:
A. True
