The tangent of two acute angles are $3$ and $2$. The sine of twice their difference is
- $\displaystyle \frac { 7 }{ 24 } $
- $\displaystyle \frac { 7 }{ 48 } $
- $\displaystyle \frac { 7 }{ 50 } $
- $\displaystyle \frac { 7 }{ 25 } $
First, find the tangent of the difference of the two angles using the identity tan(A - B) = (tan A - tan B) / (1 + tan A tan B), which yields 1/7. Then, use the double-angle formula sin(2 theta) = 2 tan(theta) / (1 + tan^2(theta)) with theta = A - B to get 2(1/7) / (1 + 1/49) = 7/25.
Given tan A equals 3 and tan B equals 2, we find the cosine and sine values to compute the required expression. Cosine A is 1 divided by the square root of 10 and cosine B is 1 divided by the square root of 5. Using the double angle formula, sin of twice A minus twice B equals 2 times sin of A minus B times cos of A minus B. Since cos of A minus B equals cos A cos B plus sin A sin B, this results in 7 divided by 25.