If `sintheta+costheta=(7)/(5)`, find `sintheta.costheta`.

Answer

To solve the problem where \( \sin \theta + \cos \theta = \frac{7}{5} \) and we need to find \( \sin \theta \cdot \cos \theta \), we can follow these steps:

### Step 1: Square both sides of the equation
We start with the equation:
\[ \sin \theta + \cos \theta = \frac{7}{5} \]
Now, we square both sides:
\[ (\sin \theta + \cos \theta)^2 = \left(\frac{7}{5}\right)^2 \]

### Step 2: Expand the left-hand side
Using the identity \( (a + b)^2 = a^2 + b^2 + 2ab \), we expand the left-hand side:
\[ \sin^2 \theta + \cos^2 \theta + 2 \sin \theta \cos \theta = \frac{49}{25} \]

### Step 3: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting this into our equation gives:
\[ 1 + 2 \sin \theta \cos \theta = \frac{49}{25} \]

### Step 4: Isolate \( 2 \sin \theta \cos \theta \)
Now, we can isolate \( 2 \sin \theta \cos \theta \):
\[ 2 \sin \theta \cos \theta = \frac{49}{25} - 1 \]
To subtract 1, we convert it to a fraction with a denominator of 25:
\[ 1 = \frac{25}{25} \]
Thus, we have:
\[ 2 \sin \theta \cos \theta = \frac{49}{25} - \frac{25}{25} = \frac{24}{25} \]

### Step 5: Solve for \( \sin \theta \cos \theta \)
Now, divide both sides by 2 to find \( \sin \theta \cos \theta \):
\[ \sin \theta \cos \theta = \frac{24}{25} \cdot \frac{1}{2} = \frac{24}{50} = \frac{12}{25} \]

### Final Answer
Thus, the value of \( \sin \theta \cdot \cos \theta \) is:
\[ \sin \theta \cdot \cos \theta = \frac{12}{25} \]