If sin x cos y = 1/4 and 3 tan x = 4 tan y, Find sin(x − y)
Question:
If \[ \sin x\cos y=\frac14 \] and \[ 3\tan x=4\tan y \] then \[ \sin(x-y) \] is equal to …………………………………….
If \[ \sin x\cos y=\frac14 \] and \[ 3\tan x=4\tan y \] then \[ \sin(x-y) \] is equal to …………………………………….
Solution
Given,
\[ 3\tan x=4\tan y \]
Therefore,
\[ 3\left(\frac{\sin x}{\cos x}\right) = 4\left(\frac{\sin y}{\cos y}\right) \]
Cross multiplying,
\[ 3\sin x\cos y = 4\sin y\cos x \]
Since
\[ \sin x\cos y=\frac14 \]
we get
\[ 3\left(\frac14\right) = 4\sin y\cos x \]
\[ \sin y\cos x = \frac{3}{16} \]
Now use the identity:
\[ \sin(x-y) = \sin x\cos y-\cos x\sin y \]
Substituting values,
\[ \sin(x-y) = \frac14-\frac{3}{16} \]
\[ = \frac{4-3}{16} \]
\[ = \frac{1}{16} \]
Therefore,
\[ \boxed{\frac{1}{16}} \]