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To simplify and solve the expression [tex]\(\sqrt{\frac{1 + \cos A}{1 - \cos A}}\)[/tex], we will use trigonometric identities and step-by-step manipulation. Here’s a comprehensive breakdown:
### Step 1: Recall Trigonometric Identities
To simplify the given expression, let's use the following trigonometric identities:
1. [tex]\(\cos A = 1 - 2\sin^2\left(\frac{A}{2}\right)\)[/tex]
2. [tex]\(1 - \cos A = 2\sin^2\left(\frac{A}{2}\right)\)[/tex]
3. [tex]\(1 + \cos A = 2\cos^2\left(\frac{A}{2}\right)\)[/tex]
### Step 2: Apply the Identities
Rewrite the components of the fraction using the identities:
[tex]\[ 1 + \cos A = 2\cos^2\left(\frac{A}{2}\right) \][/tex]
[tex]\[ 1 - \cos A = 2\sin^2\left(\frac{A}{2}\right) \][/tex]
### Step 3: Substitute the Identities
Now, substitute these into our original expression:
[tex]\[ \sqrt{\frac{1 + \cos A}{1 - \cos A}} = \sqrt{\frac{2\cos^2\left(\frac{A}{2}\right)}{2\sin^2\left(\frac{A}{2}\right)}} \][/tex]
### Step 4: Simplify the Fraction
Simplify the fraction inside the square root:
[tex]\[ \sqrt{\frac{2\cos^2\left(\frac{A}{2}\right)}{2\sin^2\left(\frac{A}{2}\right)}} = \sqrt{\frac{\cos^2\left(\frac{A}{2}\right)}{\sin^2\left(\frac{A}{2}\right)}} \][/tex]
### Step 5: Simplify the Expression Further
This can be further simplified because the fraction of squares can be reduced:
[tex]\[ \sqrt{\frac{\cos^2\left(\frac{A}{2}\right)}{\sin^2\left(\frac{A}{2}\right)}} = \sqrt{\left(\frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)}\right)^2} \][/tex]
### Step 6: Take the Square Root
Now, taking the square root of a square:
[tex]\[ \sqrt{\left(\frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)}\right)^2} = \left| \frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)} \right| \][/tex]
Since [tex]\(\frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)}\)[/tex] is the cotangent of [tex]\(\frac{A}{2}\)[/tex], we have:
[tex]\[ \left| \frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)} \right| = \left| \cot \left(\frac{A}{2}\right) \right| \][/tex]
### Final Answer
Thus, the expression simplifies to:
[tex]\[ \sqrt{\frac{1 + \cos A}{1 - \cos A}} = \left| \cot \left(\frac{A}{2}\right) \right| \][/tex]
The absolute value notation is due to the fact that square roots yield non-negative results, while the cotangent function can be either positive or negative depending on the angle.
### Step 1: Recall Trigonometric Identities
To simplify the given expression, let's use the following trigonometric identities:
1. [tex]\(\cos A = 1 - 2\sin^2\left(\frac{A}{2}\right)\)[/tex]
2. [tex]\(1 - \cos A = 2\sin^2\left(\frac{A}{2}\right)\)[/tex]
3. [tex]\(1 + \cos A = 2\cos^2\left(\frac{A}{2}\right)\)[/tex]
### Step 2: Apply the Identities
Rewrite the components of the fraction using the identities:
[tex]\[ 1 + \cos A = 2\cos^2\left(\frac{A}{2}\right) \][/tex]
[tex]\[ 1 - \cos A = 2\sin^2\left(\frac{A}{2}\right) \][/tex]
### Step 3: Substitute the Identities
Now, substitute these into our original expression:
[tex]\[ \sqrt{\frac{1 + \cos A}{1 - \cos A}} = \sqrt{\frac{2\cos^2\left(\frac{A}{2}\right)}{2\sin^2\left(\frac{A}{2}\right)}} \][/tex]
### Step 4: Simplify the Fraction
Simplify the fraction inside the square root:
[tex]\[ \sqrt{\frac{2\cos^2\left(\frac{A}{2}\right)}{2\sin^2\left(\frac{A}{2}\right)}} = \sqrt{\frac{\cos^2\left(\frac{A}{2}\right)}{\sin^2\left(\frac{A}{2}\right)}} \][/tex]
### Step 5: Simplify the Expression Further
This can be further simplified because the fraction of squares can be reduced:
[tex]\[ \sqrt{\frac{\cos^2\left(\frac{A}{2}\right)}{\sin^2\left(\frac{A}{2}\right)}} = \sqrt{\left(\frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)}\right)^2} \][/tex]
### Step 6: Take the Square Root
Now, taking the square root of a square:
[tex]\[ \sqrt{\left(\frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)}\right)^2} = \left| \frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)} \right| \][/tex]
Since [tex]\(\frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)}\)[/tex] is the cotangent of [tex]\(\frac{A}{2}\)[/tex], we have:
[tex]\[ \left| \frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)} \right| = \left| \cot \left(\frac{A}{2}\right) \right| \][/tex]
### Final Answer
Thus, the expression simplifies to:
[tex]\[ \sqrt{\frac{1 + \cos A}{1 - \cos A}} = \left| \cot \left(\frac{A}{2}\right) \right| \][/tex]
The absolute value notation is due to the fact that square roots yield non-negative results, while the cotangent function can be either positive or negative depending on the angle.
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