If `sectheta+tantheta=5`, then find `sintheta=?`

Answer

To solve the equation \( \sec \theta + \tan \theta = 5 \) and find \( \sin \theta \), we can follow these steps:

### Step 1: Use the Pythagorean Identity
We know from trigonometric identities that:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]
Let \( x = \sec \theta \) and \( y = \tan \theta \). Then we have:
\[ x + y = 5 \]
and
\[ x^2 - y^2 = 1 \]

### Step 2: Express \( y \) in terms of \( x \)
From the first equation:
\[ y = 5 - x \]

### Step 3: Substitute \( y \) into the second equation
Substituting \( y \) in the second equation gives:
\[ x^2 - (5 - x)^2 = 1 \]

### Step 4: Expand and simplify
Expanding \( (5 - x)^2 \):
\[ x^2 - (25 - 10x + x^2) = 1 \]
This simplifies to:
\[ x^2 - 25 + 10x - x^2 = 1 \]
which reduces to:
\[ 10x - 25 = 1 \]

### Step 5: Solve for \( x \)
Adding 25 to both sides:
\[ 10x = 26 \]
Dividing by 10:
\[ x = \frac{26}{10} = \frac{13}{5} \]

### Step 6: Find \( y \)
Now substitute \( x \) back to find \( y \):
\[ y = 5 - x = 5 - \frac{13}{5} = \frac{25}{5} - \frac{13}{5} = \frac{12}{5} \]

### Step 7: Use \( \sec \theta \) and \( \tan \theta \) to find \( \sin \theta \)
We know:
\[ \sec \theta = \frac{1}{\cos \theta} = \frac{13}{5} \implies \cos \theta = \frac{5}{13} \]
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{12}{5} \implies \sin \theta = \tan \theta \cdot \cos \theta = \frac{12}{5} \cdot \frac{5}{13} = \frac{12}{13} \]

### Step 8: Conclusion
Thus, the value of \( \sin \theta \) is:
\[ \sin \theta = \frac{12}{13} \]