If `0 lt theta_(2) lt theta_(1) lt pi/4, cos (theta_(1) + theta_(2)) = 3/5` and `cos(theta_(1)-theta_(2))=4/5`, then `sintheta_(1) sintheta_(2)` equal to

Options

A. `-1`
B. 1
C. 2
D. `-2 `

Answer

To solve the problem, we will use the given trigonometric identities and manipulate them step by step.

### Step 1: Write down the given equations
We have two equations based on the problem statement:
1. \( \cos(\theta_1 + \theta_2) = \frac{3}{5} \)
2. \( \cos(\theta_1 - \theta_2) = \frac{4}{5} \)

### Step 2: Use the cosine addition and subtraction formulas
Using the cosine addition and subtraction formulas, we can rewrite these equations as:
1. \( \cos(\theta_1 + \theta_2) = \cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2 = \frac{3}{5} \) (Equation 1)
2. \( \cos(\theta_1 - \theta_2) = \cos \theta_1 \cos \theta_2 + \sin \theta_1 \sin \theta_2 = \frac{4}{5} \) (Equation 2)

### Step 3: Set up the equations
Let:
- \( a = \cos \theta_1 \cos \theta_2 \)
- \( b = \sin \theta_1 \sin \theta_2 \)

Then we can rewrite the equations as:
1. \( a - b = \frac{3}{5} \) (Equation 1)
2. \( a + b = \frac{4}{5} \) (Equation 2)

### Step 4: Solve for \( a \) and \( b \)
Now, we can add and subtract these two equations to find \( a \) and \( b \).

**Adding the equations:**
\[ (a - b) + (a + b) = \frac{3}{5} + \frac{4}{5} \]
\[ 2a = \frac{7}{5} \implies a = \frac{7}{10} \]

**Subtracting the equations:**
\[ (a + b) - (a - b) = \frac{4}{5} - \frac{3}{5} \]
\[ 2b = \frac{1}{5} \implies b = \frac{1}{10} \]

### Step 5: Relate \( b \) back to \( \sin \theta_1 \sin \theta_2 \)
Since \( b = \sin \theta_1 \sin \theta_2 \), we have:
\[ \sin \theta_1 \sin \theta_2 = \frac{1}{10} \]

### Final Answer
Thus, the value of \( \sin \theta_1 \sin \theta_2 \) is \( \frac{1}{10} \).

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