From simple questions to complex issues, IDNLearn.com has the answers you need. Discover prompt and accurate answers from our experts, ensuring you get the information you need quickly.
Sagot :
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]
### 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]
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! Your search for solutions ends at IDNLearn.com. Thank you for visiting, and we look forward to helping you again.
