If `sectheta - tantheta =3 ` then `costheta` is equal to :

Options

A. `(4)/(9)`
B. `(3)/(7)`
C. `(2)/(5)`
D. `(3)/(5)`

Answer

To solve the equation \( \sec \theta - \tan \theta = 3 \) and find \( \cos \theta \), we can follow these steps:

### Step 1: Use the identity for secant and tangent
We know that:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]
This can be factored as:
\[ (\sec \theta - \tan \theta)(\sec \theta + \tan \theta) = 1 \]

### Step 2: Substitute the given value
From the problem, we have:
\[ \sec \theta - \tan \theta = 3 \]
Let’s denote \( x = \sec \theta + \tan \theta \). Then we can write:
\[ 3x = 1 \implies x = \frac{1}{3} \]
So, we have:
\[ \sec \theta + \tan \theta = \frac{1}{3} \]

### Step 3: Set up the equations
Now we have two equations:
1. \( \sec \theta - \tan \theta = 3 \)
2. \( \sec \theta + \tan \theta = \frac{1}{3} \)

### Step 4: Add the equations
Adding these two equations:
\[ (\sec \theta - \tan \theta) + (\sec \theta + \tan \theta) = 3 + \frac{1}{3} \]
This simplifies to:
\[ 2 \sec \theta = 3 + \frac{1}{3} \]
Calculating the right-hand side:
\[ 3 + \frac{1}{3} = \frac{9}{3} + \frac{1}{3} = \frac{10}{3} \]
Thus:
\[ 2 \sec \theta = \frac{10}{3} \]

### Step 5: Solve for sec theta
Dividing both sides by 2:
\[ \sec \theta = \frac{10}{3} \cdot \frac{1}{2} = \frac{5}{3} \]

### Step 6: Find cos theta
Since \( \sec \theta = \frac{1}{\cos \theta} \), we can write:
\[ \frac{1}{\cos \theta} = \frac{5}{3} \]
Taking the reciprocal gives:
\[ \cos \theta = \frac{3}{5} \]

### Final Answer
Thus, the value of \( \cos \theta \) is:
\[ \cos \theta = \frac{3}{5} \]

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