Answered

Verify the identity:

[tex]\[ \frac{\cos (u) \sec (u)}{\tan (u)} = \cot (u) \][/tex]

Use a reciprocal identity to rewrite the expression, and then simplify:

[tex]\[
\begin{aligned}
\frac{\cos (u) \sec (u)}{\tan (u)} & = \left(\cos (u) \cdot \sec (u)\right) \left(\frac{1}{\tan (u)}\right) \\
& = \cos (u) \cdot \frac{1}{\cos (u)} \cdot \cot (u) \\
& = \cot (u)
\end{aligned}
\][/tex]



Answer :

GinnyAnswer
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]

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