Certainly! Let's solve the given expression step-by-step: [tex]\(\frac{\cot^2 x}{7 + \cot x}\)[/tex].
### Step 1: Recall the definition of cotangent
The cotangent of [tex]\(x\)[/tex] is defined as:
[tex]\[ \cot(x) = \frac{1}{\tan(x)} \][/tex]
### Step 2: Simplify [tex]\(\cot^2(x)\)[/tex]
Using the definition of cotangent, we can express [tex]\(\cot^2(x)\)[/tex] in terms of tangent:
[tex]\[ \cot^2(x) = \left(\frac{1}{\tan(x)}\right)^2 = \frac{1}{\tan^2(x)} \][/tex]
### Step 3: Substitute [tex]\(\cot(x)\)[/tex] in the denominator
Now, let’s rewrite the given expression with these substitutions:
[tex]\[ \frac{\frac{1}{\tan^2(x)}}{7 + \frac{1}{\tan(x)}} \][/tex]
### Step 4: Simplify the denominator
To simplify the denominator, we need a common denominator:
[tex]\[ 7 + \frac{1}{\tan(x)} = \frac{7\tan(x) + 1}{\tan(x)} \][/tex]
### Step 5: Combine the fractions
Now that we have a common denominator, substitute back into the main fraction:
[tex]\[ \frac{\frac{1}{\tan^2(x)}}{\frac{7\tan(x) + 1}{\tan(x)}} \][/tex]
### Step 6: Simplify the complex fraction
When dividing by a fraction, we multiply by its reciprocal:
[tex]\[ \frac{1}{\tan^2(x)} \times \frac{\tan(x)}{7\tan(x) + 1} \][/tex]
### Step 7: Combine the numerator and denominator
Multiply the numerators and denominators:
[tex]\[ \frac{\tan(x)}{\tan^2(x) \times (7\tan(x) + 1)} \][/tex]
Simplify the [tex]\(\tan(x)\)[/tex] terms:
[tex]\[ \frac{1}{\tan(x) \times (7\tan(x) + 1)} \][/tex]
So, the simplified form of the given expression is:
[tex]\[ \frac{1}{\tan(x)(7\tan(x) + 1)} \][/tex]
Therefore, the final simplified expression is:
[tex]\[ \frac{1}{\tan(x)(7\tan(x) + 1)} \][/tex]
