If sin(x+y)/sin(x−y) = (a+b)/(a−b), Show that tanx/tany = a/b
Question
If
\[ \frac{\sin(x+y)}{\sin(x-y)} = \frac{a+b}{a-b} \]
show that:
\[ \frac{\tan x}{\tan y} = \frac{a}{b} \]
Proof
\[ \frac{\sin(x+y)}{\sin(x-y)} = \frac{a+b}{a-b} \]
Using
\[ \sin(x+y)=\sin x\cos y+\cos x\sin y \]
and
\[ \sin(x-y)=\sin x\cos y-\cos x\sin y \]
\[ \frac{\sin x\cos y+\cos x\sin y} {\sin x\cos y-\cos x\sin y} = \frac{a+b}{a-b} \]
Cross multiplying,
\[ (a-b)(\sin x\cos y+\cos x\sin y) \]
\[ = (a+b)(\sin x\cos y-\cos x\sin y) \]
\[ a\sin x\cos y-b\sin x\cos y +a\cos x\sin y-b\cos x\sin y \]
\[ = a\sin x\cos y+b\sin x\cos y -a\cos x\sin y-b\cos x\sin y \]
\[ -2b\sin x\cos y = -2a\cos x\sin y \]
\[ b\sin x\cos y = a\cos x\sin y \]
\[ \frac{\sin x}{\cos x} = \frac{a\sin y}{b\cos y} \]
\[ \tan x = \frac{a}{b}\tan y \]
\[ \frac{\tan x}{\tan y} = \frac{a}{b} \]
Hence proved.