Express `sectheta` in terms of `cottheta`

Answer

To express \( \sec \theta \) in terms of \( \cot \theta \), we can follow these steps:

### Step 1: Use the identity for \( \sec^2 \theta \)
We know from trigonometric identities that:
\[ \sec^2 \theta = 1 + \tan^2 \theta \]

### Step 2: Express \( \tan \theta \) in terms of \( \cot \theta \)
Since \( \tan \theta \) is the reciprocal of \( \cot \theta \), we have:
\[ \tan \theta = \frac{1}{\cot \theta} \]

### Step 3: Substitute \( \tan^2 \theta \) into the identity
Now, substituting \( \tan \theta \) into the identity:
\[ \tan^2 \theta = \left(\frac{1}{\cot \theta}\right)^2 = \frac{1}{\cot^2 \theta} \]
Thus, we can rewrite the identity as:
\[ \sec^2 \theta = 1 + \frac{1}{\cot^2 \theta} \]

### Step 4: Combine the terms on the right side
To combine the terms, we can express \( 1 \) as \( \frac{\cot^2 \theta}{\cot^2 \theta} \):
\[ \sec^2 \theta = \frac{\cot^2 \theta + 1}{\cot^2 \theta} \]

### Step 5: Take the square root to find \( \sec \theta \)
Now, to find \( \sec \theta \), we take the square root of both sides:
\[ \sec \theta = \sqrt{\sec^2 \theta} = \sqrt{\frac{\cot^2 \theta + 1}{\cot^2 \theta}} \]

### Step 6: Simplify the expression
This can be simplified further:
\[ \sec \theta = \frac{\sqrt{\cot^2 \theta + 1}}{\sqrt{\cot^2 \theta}} \]
Since \( \sqrt{\cot^2 \theta} = \cot \theta \), we have:
\[ \sec \theta = \frac{\sqrt{\cot^2 \theta + 1}}{\cot \theta} \]

### Final Result
Thus, the expression for \( \sec \theta \) in terms of \( \cot \theta \) is:
\[ \sec \theta = \frac{\sqrt{\cot^2 \theta + 1}}{\cot \theta} \]

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