`cos theta sqrt(sec^2 theta-1)` is equal to

Options

A. sin `theta`
B. cot `theta`
C. sec `theta` s
D. 1

Answer

To solve the expression \( \cos \theta \sqrt{\sec^2 \theta - 1} \), we can follow these steps:

### Step 1: Understand the identity
We know that \( \sec^2 \theta = 1 + \tan^2 \theta \). This is a fundamental trigonometric identity.

### Step 2: Substitute the identity
Using the identity from Step 1, we can rewrite the expression inside the square root:
\[ \sec^2 \theta - 1 = \tan^2 \theta \]

### Step 3: Substitute back into the expression
Now, substitute \( \tan^2 \theta \) back into the original expression:
\[ \sqrt{\sec^2 \theta - 1} = \sqrt{\tan^2 \theta} \]

### Step 4: Simplify the square root
The square root of \( \tan^2 \theta \) is \( |\tan \theta| \). However, since we are dealing with angles in the context of trigonometric functions, we can simplify this to:
\[ \sqrt{\tan^2 \theta} = \tan \theta \]

### Step 5: Substitute back into the expression
Now, substitute \( \tan \theta \) back into the expression:
\[ \cos \theta \cdot \tan \theta \]

### Step 6: Use the definition of tangent
Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Therefore, we can rewrite the expression:
\[ \cos \theta \cdot \tan \theta = \cos \theta \cdot \frac{\sin \theta}{\cos \theta} \]

### Step 7: Simplify the expression
The \( \cos \theta \) in the numerator and denominator cancels out:
\[ \cos \theta \cdot \frac{\sin \theta}{\cos \theta} = \sin \theta \]

### Final Answer
Thus, the expression \( \cos \theta \sqrt{\sec^2 \theta - 1} \) simplifies to:
\[ \sin \theta \]