If `theta` is a positive acute angle and `3(sec^(2)theta+tan^(2)theta)=5` , then the value of `cos2theta` is

Options

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

Answer

To solve the equation \(3(\sec^2 \theta + \tan^2 \theta) = 5\) and find the value of \(\cos 2\theta\), we can follow these steps:

### Step 1: Simplify the given equation
We start with the equation:
\[ 3(\sec^2 \theta + \tan^2 \theta) = 5 \]
Dividing both sides by 3 gives:
\[ \sec^2 \theta + \tan^2 \theta = \frac{5}{3} \]

**Hint:** Remember that \(\sec^2 \theta = 1 + \tan^2 \theta\).

### Step 2: Substitute \(\sec^2 \theta\)
Using the identity \(\sec^2 \theta = 1 + \tan^2 \theta\), we can rewrite the equation:
\[ 1 + \tan^2 \theta + \tan^2 \theta = \frac{5}{3} \]
This simplifies to:
\[ 1 + 2\tan^2 \theta = \frac{5}{3} \]

**Hint:** Isolate \(2\tan^2 \theta\) by subtracting 1 from both sides.

### Step 3: Solve for \(\tan^2 \theta\)
Subtracting 1 (which is \(\frac{3}{3}\)) from both sides gives:
\[ 2\tan^2 \theta = \frac{5}{3} - \frac{3}{3} = \frac{2}{3} \]
Now, divide both sides by 2:
\[ \tan^2 \theta = \frac{2}{6} = \frac{1}{3} \]

**Hint:** To find \(\tan \theta\), take the square root of both sides.

### Step 4: Find \(\tan \theta\)
Taking the square root gives:
\[ \tan \theta = \sqrt{\frac{1}{3}} = \frac{1}{\sqrt{3}} \]
Since \(\theta\) is a positive acute angle, we only consider the positive root.

**Hint:** Recall that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

### Step 5: Find \(\cos 2\theta\)
Using the double angle formula for cosine:
\[ \cos 2\theta = \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} \]
Substituting \(\tan^2 \theta = \frac{1}{3}\):
\[ \cos 2\theta = \frac{1 - \frac{1}{3}}{1 + \frac{1}{3}} = \frac{\frac{2}{3}}{\frac{4}{3}} = \frac{2}{4} = \frac{1}{2} \]

**Hint:** Check if the fraction simplifies correctly.

### Final Answer
Thus, the value of \(\cos 2\theta\) is:
\[ \cos 2\theta = \frac{1}{2} \]