Question

sin (x + y)/sin(x-y)=a+b/a-b then what is the value of tanx/tany

Answer 1

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Answer 2

$\textbf{Answer}$

Given trigonometric equation is,

$\textbf{sin(x+y) / sin(x-y) = (a+b) / (a-b)}$

We know the componendo and dividendo property, that is,
$\textbf{if a/b = c/d,}$

$\textbf{then (a+b) / (a-b) = (c+d) / (c-d)}$

So by applying this componendo and dividendo property in the given trigonometric equation,

{sin(x+y) + sin(x-y)} / {sin(x+y) - sin(x-y)} = (a+b+a-b)/a+b-a+b) -------(1)

We know that,
$\textbf{sin A+sin B = 2 sin(A+B)/2 . cos(A-B)/2}$
&
$\textbf{sin A-sin B = 2cos(A+B)/2 . sin(A-B)/2}$

Using above formula in the given equation (1),

{2 sin (x+y+x-y)/2 . cos (x+y-x+y)/2} / {2 cos (x+y+x-y)/2 . sin (x+y-x+y)/2} = 2a / 2b

=> 2 sin x . cos y / 2 cos x . sin y = a/b

=> (sin x / cos x)(cos y / sin y) = a/b

=> tan x . cot y = a/b

We know that cot y = 1 / tan y

=> tan x / tan y = a/b

$\textbf{So the value of tanx/tany is a/b}$

$\textbf{Hope It Helps}$