Question

If $\sin x\cos y=\frac{1}{4}$ and 3 tan x = 4 tan y, then sin (x – y) is equal to __________.

Answer 1

$$\begin{gathered} \mathrm{Given}\sin x\cos y=\frac{1}{4} \\ \mathrm{and}3\tan x=4\tan y\mathrm{i}.\mathrm{e}.\tan x=\frac{4}{3}\tan y \\ \mathrm{i}.\mathrm{e}.\frac{\tan x}{\tan y}=\frac{4}{3} \\ \mathrm{i}.\mathrm{e}.\frac{\sin x}{\cos x}\frac{\cos y}{\sin y}=\frac{4}{3} \\ \Rightarrow \frac{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}{\cos x\sin y}=\frac{4}{3} \\ \Rightarrow \cos x\sin y=\frac{3}{16} \\ \therefore \sin \left(x-y\right)=\sin x\cos y-\cos x\sin y \\ =\frac{1}{4}-\frac{3}{16} \\ =\frac{4-3}{16} \\ \sin \left(x-y\right)=\frac{1}{16} \\ \mathrm{hence},\mathrm{value}\mathrm{of}\sin \left(x-y\right)=\frac{1}{16}. \end{gathered}$$