If ` sectheta+tantheta=sqrt(2)`,find the value of `sintheta`.

Options

A. `(1)/(2)`
B. `(1)/(4)`
C. `(1)/(sqrt(2))`
D. `(1)/(3)`

Answer

To solve the equation \( \sec \theta + \tan \theta = \sqrt{2} \) and find the value of \( \sin \theta \), we can follow these steps:

### Step 1: Rewrite the equation using definitions
We know that:
\[ \sec \theta = \frac{1}{\cos \theta} \quad \text{and} \quad \tan \theta = \frac{\sin \theta}{\cos \theta} \]
Substituting these into the equation gives:
\[ \frac{1}{\cos \theta} + \frac{\sin \theta}{\cos \theta} = \sqrt{2} \]

### Step 2: Combine the fractions
Combining the fractions on the left side:
\[ \frac{1 + \sin \theta}{\cos \theta} = \sqrt{2} \]

### Step 3: Cross-multiply
Cross-multiplying gives:
\[ 1 + \sin \theta = \sqrt{2} \cos \theta \]

### Step 4: Square both sides
To eliminate the square root, we square both sides:
\[ (1 + \sin \theta)^2 = (\sqrt{2} \cos \theta)^2 \]
This simplifies to:
\[ 1 + 2\sin \theta + \sin^2 \theta = 2 \cos^2 \theta \]

### Step 5: Use the Pythagorean identity
Using the identity \( \cos^2 \theta = 1 - \sin^2 \theta \), we substitute:
\[ 1 + 2\sin \theta + \sin^2 \theta = 2(1 - \sin^2 \theta) \]
This expands to:
\[ 1 + 2\sin \theta + \sin^2 \theta = 2 - 2\sin^2 \theta \]

### Step 6: Rearrange the equation
Rearranging gives:
\[ 1 + 2\sin \theta + \sin^2 \theta + 2\sin^2 \theta - 2 = 0 \]
This simplifies to:
\[ 3\sin^2 \theta + 2\sin \theta - 1 = 0 \]

### Step 7: Solve the quadratic equation
Let \( x = \sin \theta \). The equation becomes:
\[ 3x^2 + 2x - 1 = 0 \]
We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{-2 \pm \sqrt{(2)^2 - 4 \cdot 3 \cdot (-1)}}{2 \cdot 3} \]
Calculating the discriminant:
\[ x = \frac{-2 \pm \sqrt{4 + 12}}{6} = \frac{-2 \pm \sqrt{16}}{6} = \frac{-2 \pm 4}{6} \]
This gives us two solutions:
\[ x = \frac{2}{6} = \frac{1}{3} \quad \text{and} \quad x = \frac{-6}{6} = -1 \]

### Step 8: Choose the valid solution
Since \( \sin \theta \) must be between -1 and 1, we discard \( -1 \) and keep:
\[ \sin \theta = \frac{1}{3} \]

### Final Answer
Thus, the value of \( \sin \theta \) is:
\[ \boxed{\frac{1}{3}} \]