Given that `cos^(2) theta - sin^(2) theta = (3)/(4) ` What is the value of cos `theta` ?

Options

A. `(sqrt(3))/(2)`
B. `(1)/(2)`
C. `(sqrt(7))/(2)`
D. `(sqrt(7))/(sqrt(8))`

Answer

To solve the equation \( \cos^2 \theta - \sin^2 \theta = \frac{3}{4} \), we can follow these steps:

### Step 1: Use the Pythagorean Identity
We know from the Pythagorean identity that:
\[ \cos^2 \theta + \sin^2 \theta = 1 \]
From this, we can express \( \sin^2 \theta \) in terms of \( \cos^2 \theta \):
\[ \sin^2 \theta = 1 - \cos^2 \theta \]

### Step 2: Substitute into the Given Equation
Now, substitute \( \sin^2 \theta \) into the original equation:
\[ \cos^2 \theta - (1 - \cos^2 \theta) = \frac{3}{4} \]
This simplifies to:
\[ \cos^2 \theta - 1 + \cos^2 \theta = \frac{3}{4} \]
\[ 2\cos^2 \theta - 1 = \frac{3}{4} \]

### Step 3: Solve for \( \cos^2 \theta \)
Now, add 1 to both sides:
\[ 2\cos^2 \theta = \frac{3}{4} + 1 \]
Convert 1 to a fraction with a denominator of 4:
\[ 1 = \frac{4}{4} \]
So,
\[ 2\cos^2 \theta = \frac{3}{4} + \frac{4}{4} = \frac{7}{4} \]
Now, divide both sides by 2:
\[ \cos^2 \theta = \frac{7}{4} \cdot \frac{1}{2} = \frac{7}{8} \]

### Step 4: Find \( \cos \theta \)
To find \( \cos \theta \), take the square root of both sides:
\[ \cos \theta = \sqrt{\frac{7}{8}} = \frac{\sqrt{7}}{\sqrt{8}} = \frac{\sqrt{7}}{2\sqrt{2}} = \frac{\sqrt{14}}{4} \]

Thus, the value of \( \cos \theta \) is:
\[ \cos \theta = \frac{\sqrt{14}}{4} \]