the value of `sqrt(1-sin^2 11 0^(@))*sec11 0^(@)`

Options

A. 2
B. -1
C. -2
D. 1

Answer

To solve the expression \( \sqrt{1 - \sin^2 110^\circ} \cdot \sec 110^\circ \), we can follow these steps:

### Step 1: Simplify the expression inside the square root
We know from the Pythagorean identity that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Thus, we can rewrite:
\[ 1 - \sin^2 110^\circ = \cos^2 110^\circ \]

### Step 2: Substitute back into the expression
Now, substituting this back into the original expression gives us:
\[ \sqrt{1 - \sin^2 110^\circ} = \sqrt{\cos^2 110^\circ} = |\cos 110^\circ| \]
Since \( 110^\circ \) is in the second quadrant, where cosine is negative, we have:
\[ |\cos 110^\circ| = -\cos 110^\circ \]

### Step 3: Write the secant function
The secant function is defined as:
\[ \sec \theta = \frac{1}{\cos \theta} \]
Thus, we can write:
\[ \sec 110^\circ = \frac{1}{\cos 110^\circ} \]

### Step 4: Combine the results
Now we can combine our results:
\[ \sqrt{1 - \sin^2 110^\circ} \cdot \sec 110^\circ = (-\cos 110^\circ) \cdot \left(\frac{1}{\cos 110^\circ}\right) \]

### Step 5: Simplify the expression
This simplifies to:
\[ -\cos 110^\circ \cdot \frac{1}{\cos 110^\circ} = -1 \]

### Final Answer
Thus, the value of \( \sqrt{1 - \sin^2 110^\circ} \cdot \sec 110^\circ \) is:
\[ \boxed{-1} \]

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