Question

The tangent of two acute angles are $3$ and $2$. The sine of twice their difference is

• A. $\frac{7}{24}$
• B. $\frac{7}{48}$
• C. $\frac{7}{50}$
• D. $\frac{7}{25}$

Answer 1

The correct option is D $\frac{7}{25}$

Given $tanA=3$ and $tanB=2$

$A-B=ta{n}^{-1}3-ta{n}^{-1}2$

$A-B=ta{n}^{-1}\frac{3-2}{1+6}=ta{n}^{-1}\frac{1}{7}=si{n}^{-1}\frac{1}{5\sqrt{2}}$

$sin(A-B)=\frac{1}{5\sqrt{2}}$

$cos(A-B)=\frac{7}{5\sqrt{2}}$

$sin\left(2\right(A-B\left)\right)=2sin(A-B)cos(A-B)=\frac{14}{50}=\frac{7}{25}$