If `sectheta+tantheta=3`, then find the value of `sectheta`.

Options

A. `(3)/(4)`
B. `(5)/(3)`
C. `(1)/(6)`
D. 9

Answer

To solve the equation \( \sec \theta + \tan \theta = 3 \), we can follow these steps:

### Step 1: Use the Pythagorean Identity
We know from trigonometric identities that:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]
This can be rearranged to express \( \sec^2 \theta \) in terms of \( \tan^2 \theta \):
\[ \sec^2 \theta = \tan^2 \theta + 1 \]

### Step 2: Set Up the Equations
Let:
\[ x = \sec \theta \]
\[ y = \tan \theta \]
From the problem, we have:
\[ x + y = 3 \]
And from the identity:
\[ x^2 - y^2 = 1 \]

### Step 3: Express \( y \) in Terms of \( x \)
From the first equation, we can express \( y \):
\[ y = 3 - x \]

### Step 4: Substitute \( y \) into the Identity
Now, substitute \( y \) into the identity:
\[ x^2 - (3 - x)^2 = 1 \]

### Step 5: Expand and Simplify
Expanding \( (3 - x)^2 \):
\[ x^2 - (9 - 6x + x^2) = 1 \]
This simplifies to:
\[ x^2 - 9 + 6x - x^2 = 1 \]
Thus:
\[ 6x - 9 = 1 \]

### Step 6: Solve for \( x \)
Now, add 9 to both sides:
\[ 6x = 10 \]
Dividing both sides by 6 gives:
\[ x = \frac{10}{6} = \frac{5}{3} \]

### Step 7: Conclusion
Thus, the value of \( \sec \theta \) is:
\[ \sec \theta = \frac{5}{3} \]