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Sagot :
Certainly! Let's solve the equation step by step using trigonometric identities and simplifications.
Given the equation:
[tex]\[ \frac{\cos 2\theta}{1 + \sin 2\theta} = \frac{1 - \tan \theta}{1 + \tan \theta} \][/tex]
### Step-by-Step Solution:
#### 1. Simplify the Left-Hand Side (LHS):
The left-hand side of the equation is:
[tex]\[ \frac{\cos 2\theta}{1 + \sin 2\theta} \][/tex]
We use the double angle identities:
[tex]\[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta \][/tex]
[tex]\[ \sin 2\theta = 2 \sin \theta \cos \theta \][/tex]
Substitute these into the LHS:
[tex]\[ \frac{\cos^2 \theta - \sin^2 \theta}{1 + 2 \sin \theta \cos \theta} \][/tex]
#### 2. Simplify the Right-Hand Side (RHS):
The right-hand side of the equation is:
[tex]\[ \frac{1 - \tan \theta}{1 + \tan \theta} \][/tex]
Recall that [tex]\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)[/tex]. Substitute this into the RHS:
[tex]\[ \frac{1 - \frac{\sin \theta}{\cos \theta}}{1 + \frac{\sin \theta}{\cos \theta}} \][/tex]
Combine the fractions:
[tex]\[ \frac{\frac{\cos \theta - \sin \theta}{\cos \theta}}{\frac{\cos \theta + \sin \theta}{\cos \theta}} \][/tex]
Simplify the expression by multiplying both the numerator and the denominator by [tex]\(\cos \theta\)[/tex]:
[tex]\[ \frac{\cos \theta - \sin \theta}{\cos \theta + \sin \theta} \][/tex]
So the RHS is:
[tex]\[ \frac{\cos \theta - \sin \theta}{\cos \theta + \sin \theta} \][/tex]
### 3. Compare the Simplified LHS and RHS:
We now have:
[tex]\[ \frac{\cos^2 \theta - \sin^2 \theta}{1 + 2 \sin \theta \cos \theta} \quad \text{and} \quad \frac{\cos \theta - \sin \theta}{\cos \theta + \sin \theta} \][/tex]
Next, we observe that:
[tex]\[ \cos^2 \theta - \sin^2 \theta = (\cos \theta - \sin \theta)(\cos \theta + \sin \theta) \][/tex]
[tex]\[ 1 + 2 \sin \theta \cos \theta = (\cos \theta + \sin \theta)^2 \][/tex]
So we substitute back into the LHS:
[tex]\[ \frac{(\cos \theta - \sin \theta)(\cos \theta + \sin \theta)}{(\cos \theta + \sin \theta)^2} \][/tex]
Simplify the fraction by canceling [tex]\((\cos \theta + \sin \theta)\)[/tex] in the numerator and the denominator:
[tex]\[ \frac{\cos \theta - \sin \theta}{\cos \theta + \sin \theta} \][/tex]
### 4. Conclusion:
Both the simplified left-hand side and right-hand side expressions match:
[tex]\[ \frac{\cos 2\theta}{1 + \sin 2\theta} = \frac{\cos \theta - \sin \theta}{\cos \theta + \sin \theta} \][/tex]
[tex]\[ \frac{1 - \tan \theta}{1 + \tan \theta} = \frac{\cos \theta - \sin \theta}{\cos \theta + \sin \theta} \][/tex]
Therefore, we have verified that:
[tex]\[ \frac{\cos 2\theta}{1 + \sin 2\theta} = \frac{1 - \tan \theta}{1 + \tan \theta} \][/tex]
is indeed true.
Therefore, the equation holds.
Given the equation:
[tex]\[ \frac{\cos 2\theta}{1 + \sin 2\theta} = \frac{1 - \tan \theta}{1 + \tan \theta} \][/tex]
### Step-by-Step Solution:
#### 1. Simplify the Left-Hand Side (LHS):
The left-hand side of the equation is:
[tex]\[ \frac{\cos 2\theta}{1 + \sin 2\theta} \][/tex]
We use the double angle identities:
[tex]\[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta \][/tex]
[tex]\[ \sin 2\theta = 2 \sin \theta \cos \theta \][/tex]
Substitute these into the LHS:
[tex]\[ \frac{\cos^2 \theta - \sin^2 \theta}{1 + 2 \sin \theta \cos \theta} \][/tex]
#### 2. Simplify the Right-Hand Side (RHS):
The right-hand side of the equation is:
[tex]\[ \frac{1 - \tan \theta}{1 + \tan \theta} \][/tex]
Recall that [tex]\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)[/tex]. Substitute this into the RHS:
[tex]\[ \frac{1 - \frac{\sin \theta}{\cos \theta}}{1 + \frac{\sin \theta}{\cos \theta}} \][/tex]
Combine the fractions:
[tex]\[ \frac{\frac{\cos \theta - \sin \theta}{\cos \theta}}{\frac{\cos \theta + \sin \theta}{\cos \theta}} \][/tex]
Simplify the expression by multiplying both the numerator and the denominator by [tex]\(\cos \theta\)[/tex]:
[tex]\[ \frac{\cos \theta - \sin \theta}{\cos \theta + \sin \theta} \][/tex]
So the RHS is:
[tex]\[ \frac{\cos \theta - \sin \theta}{\cos \theta + \sin \theta} \][/tex]
### 3. Compare the Simplified LHS and RHS:
We now have:
[tex]\[ \frac{\cos^2 \theta - \sin^2 \theta}{1 + 2 \sin \theta \cos \theta} \quad \text{and} \quad \frac{\cos \theta - \sin \theta}{\cos \theta + \sin \theta} \][/tex]
Next, we observe that:
[tex]\[ \cos^2 \theta - \sin^2 \theta = (\cos \theta - \sin \theta)(\cos \theta + \sin \theta) \][/tex]
[tex]\[ 1 + 2 \sin \theta \cos \theta = (\cos \theta + \sin \theta)^2 \][/tex]
So we substitute back into the LHS:
[tex]\[ \frac{(\cos \theta - \sin \theta)(\cos \theta + \sin \theta)}{(\cos \theta + \sin \theta)^2} \][/tex]
Simplify the fraction by canceling [tex]\((\cos \theta + \sin \theta)\)[/tex] in the numerator and the denominator:
[tex]\[ \frac{\cos \theta - \sin \theta}{\cos \theta + \sin \theta} \][/tex]
### 4. Conclusion:
Both the simplified left-hand side and right-hand side expressions match:
[tex]\[ \frac{\cos 2\theta}{1 + \sin 2\theta} = \frac{\cos \theta - \sin \theta}{\cos \theta + \sin \theta} \][/tex]
[tex]\[ \frac{1 - \tan \theta}{1 + \tan \theta} = \frac{\cos \theta - \sin \theta}{\cos \theta + \sin \theta} \][/tex]
Therefore, we have verified that:
[tex]\[ \frac{\cos 2\theta}{1 + \sin 2\theta} = \frac{1 - \tan \theta}{1 + \tan \theta} \][/tex]
is indeed true.
Therefore, the equation holds.
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