To verify the identity [tex]\(\frac{\cos(u) \sec(u)}{\tan(u)} = \cot(u)\)[/tex], let's break it down step-by-step using reciprocal identities and simplification.
### Step 1: Simplify [tex]\(\cos(u) \sec(u)\)[/tex]
First, recall the reciprocal identity:
[tex]\[ \sec(u) = \frac{1}{\cos(u)} \][/tex]
Using this identity within the expression [tex]\(\cos(u) \sec(u)\)[/tex], we get:
[tex]\[ \cos(u) \sec(u) = \cos(u) \cdot \frac{1}{\cos(u)} \][/tex]
This simplifies to:
[tex]\[ \cos(u) \cdot \frac{1}{\cos(u)} = 1 \][/tex]
### Step 2: Rewrite the original expression
Now, substitute this simplification back into the original expression:
[tex]\[ \frac{\cos(u) \sec(u)}{\tan(u)} = \frac{1}{\tan(u)} \][/tex]
### Step 3: Simplify [tex]\(\frac{1}{\tan(u)}\)[/tex]
Recall the definition of tangent and its reciprocal identity:
[tex]\[ \tan(u) = \frac{\sin(u)}{\cos(u)} \][/tex]
[tex]\[ \cot(u) = \frac{1}{\tan(u)} = \frac{\cos(u)}{\sin(u)} \][/tex]
Thus,
[tex]\[ \frac{1}{\tan(u)} = \cot(u) \][/tex]
### Step 4: Combine all steps
Putting it all together:
[tex]\[ \frac{\cos(u) \sec(u)}{\tan(u)} = 1 \div \tan(u) = \frac{1}{\tan(u)} = \cot(u) \][/tex]
Therefore, we have verified the identity:
[tex]\[
\frac{\cos(u) \sec(u)}{\tan(u)} = \cot(u)
\][/tex]
