If cos(A − B) = 3/5 and tan A tan B = 2, Find sin A sin B

Solution

Using the identity:

\[ \tan A\tan B = \frac{\sin A\sin B}{\cos A\cos B} \]

Given,

\[ \tan A\tan B=2 \]

Therefore,

\[ \sin A\sin B = 2\cos A\cos B \]

Now use:

\[ \cos(A-B) = \cos A\cos B+\sin A\sin B \]

Substituting,

\[ \frac{3}{5} = \cos A\cos B+2\cos A\cos B \]

\[ \frac{3}{5} = 3\cos A\cos B \]

Therefore,

\[ \cos A\cos B = \frac{1}{5} \]

Hence,

\[ \sin A\sin B = 2\left(\frac{1}{5}\right) = \frac{2}{5} \]

Therefore,

\[ \boxed{\frac{2}{5}} \]