If `cosec theta - cot theta = 1/2, 0 lt theta lt pi/2, ` then `cos theta ` is equal to

Options

A. `(-3)/(5)`
B. `-(5)/(3)`
C. `5/3`
D. `3/5`

Answer

To solve the equation \( \csc \theta - \cot \theta = \frac{1}{2} \) for \( \cos \theta \), we will follow these steps:

### Step 1: Rewrite the equation in terms of sine and cosine.
We know that:
\[ \csc \theta = \frac{1}{\sin \theta} \quad \text{and} \quad \cot \theta = \frac{\cos \theta}{\sin \theta} \]
Substituting these into the equation gives:
\[ \frac{1}{\sin \theta} - \frac{\cos \theta}{\sin \theta} = \frac{1}{2} \]
This simplifies to:
\[ \frac{1 - \cos \theta}{\sin \theta} = \frac{1}{2} \]

### Step 2: Cross-multiply to eliminate the fraction.
Cross-multiplying gives:
\[ 2(1 - \cos \theta) = \sin \theta \]

### Step 3: Use the Pythagorean identity.
We know that \( \sin^2 \theta + \cos^2 \theta = 1 \). Therefore, we can express \( \sin \theta \) in terms of \( \cos \theta \):
\[ \sin \theta = \sqrt{1 - \cos^2 \theta} \]
Substituting this into the equation from Step 2 gives:
\[ 2(1 - \cos \theta) = \sqrt{1 - \cos^2 \theta} \]

### Step 4: Square both sides to eliminate the square root.
Squaring both sides results in:
\[ 4(1 - \cos \theta)^2 = 1 - \cos^2 \theta \]
Expanding the left side:
\[ 4(1 - 2\cos \theta + \cos^2 \theta) = 1 - \cos^2 \theta \]
This simplifies to:
\[ 4 - 8\cos \theta + 4\cos^2 \theta = 1 - \cos^2 \theta \]

### Step 5: Rearrange the equation.
Bringing all terms to one side gives:
\[ 5\cos^2 \theta - 8\cos \theta + 3 = 0 \]

### Step 6: Solve the quadratic equation.
Using the quadratic formula \( \cos \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ a = 5, \quad b = -8, \quad c = 3 \]
Calculating the discriminant:
\[ b^2 - 4ac = (-8)^2 - 4 \cdot 5 \cdot 3 = 64 - 60 = 4 \]
Now substituting into the quadratic formula:
\[ \cos \theta = \frac{8 \pm \sqrt{4}}{2 \cdot 5} = \frac{8 \pm 2}{10} \]
This gives us two possible solutions:
\[ \cos \theta = \frac{10}{10} = 1 \quad \text{or} \quad \cos \theta = \frac{6}{10} = \frac{3}{5} \]

### Step 7: Determine the valid solution.
Since \( 0 < \theta < \frac{\pi}{2} \), \( \cos \theta \) must be positive. Thus, we discard \( \cos \theta = 1 \) as it corresponds to \( \theta = 0 \), which is not in the range given. Therefore, the valid solution is:
\[ \cos \theta = \frac{3}{5} \]

### Final Answer:
\[ \cos \theta = \frac{3}{5} \]