The value of `sin (2 tan ^(-1) "" (1)/(3)) + cos (tan ^(-1) 2 sqrt2) is (p)/(q),` then pq is

Answer

To solve the problem, we need to evaluate the expression \( \sin(2 \tan^{-1}(1/3)) + \cos(\tan^{-1}(2\sqrt{2})) \).

### Step 1: Evaluate \( \cos(\tan^{-1}(2\sqrt{2})) \)

Let \( \theta = \tan^{-1}(2\sqrt{2}) \). Then, by definition of tangent:

\[ \tan(\theta) = 2\sqrt{2} = \frac{\text{opposite}}{\text{adjacent}} = \frac{2\sqrt{2}}{1} \]

We can form a right triangle where the opposite side is \( 2\sqrt{2} \) and the adjacent side is \( 1 \). To find the hypotenuse \( h \):

\[ h = \sqrt{(2\sqrt{2})^2 + 1^2} = \sqrt{8 + 1} = \sqrt{9} = 3 \]

Now, we can find \( \cos(\theta) \):

\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{1}{3} \]

### Step 2: Evaluate \( \sin(2 \tan^{-1}(1/3)) \)

Let \( \phi = \tan^{-1}(1/3) \). Then:

\[ \tan(\phi) = \frac{1}{3} = \frac{1}{3} \]

Again, we can form a right triangle where the opposite side is \( 1 \) and the adjacent side is \( 3 \). The hypotenuse \( h \) is:

\[ h = \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10} \]

Now, we can find \( \sin(\phi) \) and \( \cos(\phi) \):

\[ \sin(\phi) = \frac{1}{\sqrt{10}}, \quad \cos(\phi) = \frac{3}{\sqrt{10}} \]

Using the double angle formula for sine:

\[ \sin(2\phi) = 2 \sin(\phi) \cos(\phi) = 2 \left(\frac{1}{\sqrt{10}}\right) \left(\frac{3}{\sqrt{10}}\right) = \frac{6}{10} = \frac{3}{5} \]

### Step 3: Combine the results

Now we can combine the two results:

\[ \sin(2 \tan^{-1}(1/3)) + \cos(\tan^{-1}(2\sqrt{2})) = \frac{3}{5} + \frac{1}{3} \]

To add these fractions, we need a common denominator. The least common multiple of \( 5 \) and \( 3 \) is \( 15 \):

\[ \frac{3}{5} = \frac{9}{15}, \quad \frac{1}{3} = \frac{5}{15} \]

Now, adding them:

\[ \frac{9}{15} + \frac{5}{15} = \frac{14}{15} \]

### Step 4: Determine \( p \) and \( q \)

From the expression \( \frac{14}{15} \), we have \( p = 14 \) and \( q = 15 \).

### Step 5: Calculate \( pq \)

Now, we calculate \( pq \):

\[ pq = 14 \times 15 = 210 \]

### Final Answer

Thus, the value of \( pq \) is \( 210 \).

---