Certainly! Let's solve the given equation step by step:
### Given Equation:
[tex]\[
\frac{1 + \sin^2(x)}{1 - \sin^2(x)} = 1 + 2\tan(x)
\][/tex]
### Step 1: Simplify the Left-Hand Side
We use the Pythagorean identity:
[tex]\[
1 - \sin^2(x) = \cos^2(x)
\][/tex]
Thus, the left-hand side of the equation can be rewritten as:
[tex]\[
\frac{1 + \sin^2(x)}{\cos^2(x)}
\][/tex]
### Step 2: Simplify the Right-Hand Side
We know that:
[tex]\[
\tan(x) = \frac{\sin(x)}{\cos(x)}
\][/tex]
Therefore, the right-hand side is:
[tex]\[
1 + 2\tan(x) = 1 + 2\left(\frac{\sin(x)}{\cos(x)}\right)
\][/tex]
### Step 3: Express Both Sides with a Common Denominator
Rewrite the right-hand side with a common denominator:
[tex]\[
1 + 2\left(\frac{\sin(x)}{\cos(x)}\right) = \frac{\cos^2(x) + 2\sin(x)\cos(x)}{\cos^2(x)}
\][/tex]
So our equation becomes:
[tex]\[
\frac{1 + \sin^2(x)}{\cos^2(x)} = \frac{\cos^2(x) + 2\sin(x)\cos(x)}{\cos^2(x)}
\][/tex]
### Step 4: Set the Numerators Equal
Since the denominators are the same, set the numerators equal:
[tex]\[
1 + \sin^2(x) = \cos^2(x) + 2\sin(x)\cos(x)
\][/tex]
### Step 5: Substitute the Pythagorean Identity Again
Using the identity [tex]\(\cos^2(x) = 1 - \sin^2(x)\)[/tex], we rewrite the equation:
[tex]\[
1 + \sin^2(x) = 1 - \sin^2(x) + 2\sin(x)\cos(x)
\][/tex]
### Step 6: Collect and Simplify Like Terms
Combine all [tex]\(\sin^2(x)\)[/tex] terms on one side:
[tex]\[
1 + \sin^2(x) - 1 + \sin^2(x) = 2\sin(x)\cos(x)
\][/tex]
[tex]\[
2\sin^2(x) = 2\sin(x)\cos(x)
\][/tex]
### Step 7: Divide Both Sides by 2
[tex]\[
\sin^2(x) = \sin(x)\cos(x)
\][/tex]
### Step 8: Factor and Solve for [tex]\( \sin(x) \)[/tex]
Factoring out [tex]\(\sin(x)\)[/tex]:
[tex]\[
\sin(x)\left(\sin(x) - \cos(x)\right) = 0
\][/tex]
This gives us two solutions:
[tex]\[
\sin(x) = 0 \quad \text{or} \quad \sin(x) = \cos(x)
\][/tex]
### Step 9: Consider the Cases
1. Case 1: [tex]\(\sin(x) = 0\)[/tex]
- [tex]\(\sin(x) = 0\)[/tex] yields solutions [tex]\(x = k\pi\)[/tex] where [tex]\(k\)[/tex] is an integer.
2. Case 2: [tex]\(\sin(x) = \cos(x)\)[/tex]
- Dividing both sides by [tex]\(\cos(x)\)[/tex] (assuming [tex]\(\cos(x) \neq 0\)[/tex]), we get:
[tex]\[
\tan(x) = 1
\][/tex]
### Step 10: Solve for [tex]\(x\)[/tex]
The equation [tex]\(\tan(x) = 1\)[/tex] has solutions:
[tex]\[
x = \frac{\pi}{4} + k\pi \quad \text{where } k \text{ is an integer}
\][/tex]
### Conclusion
The general solution for the given equation is:
[tex]\[
x = \frac{\pi}{4} + k\pi \quad \text{where } k \text{ is an integer}
\][/tex]
