If `tantheta+sectheta=4`, then what is the value of `sintheta`?

Options

A. `8//17`
B. `8//15`
C. `15//17`
D. `23//32`

Explanation

To solve the equation \( \tan \theta + \sec \theta = 4 \) and find the value of \( \sin \theta \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \tan \theta + \sec \theta = 4 \]
We can express \( \tan \theta \) and \( \sec \theta \) in terms of sine and cosine:
\[ \frac{\sin \theta}{\cos \theta} + \frac{1}{\cos \theta} = 4 \]
Combining the fractions gives:
\[ \frac{\sin \theta + 1}{\cos \theta} = 4 \]

### Step 2: Cross-multiply
Cross-multiplying to eliminate the fraction, we have:
\[ \sin \theta + 1 = 4 \cos \theta \]

### Step 3: Square both sides
Next, we square both sides to eliminate the square root later on:
\[ (\sin \theta + 1)^2 = (4 \cos \theta)^2 \]
Expanding both sides:
\[ \sin^2 \theta + 2 \sin \theta + 1 = 16 \cos^2 \theta \]

### Step 4: Use the Pythagorean identity
Using the identity \( \cos^2 \theta = 1 - \sin^2 \theta \), we substitute for \( \cos^2 \theta \):
\[ \sin^2 \theta + 2 \sin \theta + 1 = 16(1 - \sin^2 \theta) \]
This simplifies to:
\[ \sin^2 \theta + 2 \sin \theta + 1 = 16 - 16 \sin^2 \theta \]

### Step 5: Rearrange the equation
Rearranging the equation gives:
\[ \sin^2 \theta + 16 \sin^2 \theta + 2 \sin \theta + 1 - 16 = 0 \]
Combining like terms results in:
\[ 17 \sin^2 \theta + 2 \sin \theta - 15 = 0 \]

### Step 6: Solve the quadratic equation
Now we can solve the quadratic equation \( 17 \sin^2 \theta + 2 \sin \theta - 15 = 0 \) using the quadratic formula:
\[ \sin \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 17 \), \( b = 2 \), and \( c = -15 \):
\[ \sin \theta = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 17 \cdot (-15)}}{2 \cdot 17} \]
Calculating the discriminant:
\[ \sin \theta = \frac{-2 \pm \sqrt{4 + 1020}}{34} \]
\[ \sin \theta = \frac{-2 \pm \sqrt{1024}}{34} \]
\[ \sin \theta = \frac{-2 \pm 32}{34} \]

### Step 7: Calculate the two possible values
Calculating the two possible values:
1. \( \sin \theta = \frac{30}{34} = \frac{15}{17} \)
2. \( \sin \theta = \frac{-34}{34} = -1 \) (not valid since sine cannot be -1 in this context)

Thus, the valid solution is:
\[ \sin \theta = \frac{15}{17} \]

### Final Answer
The value of \( \sin \theta \) is:
\[ \boxed{\frac{15}{17}} \]