Question

if tan x = tan y,then the value of sin(x+y)/sin(x-y) is​

Answer 1

Step-by-step explanation:

Given trigonometric equation is,

\textbf{sin(x+y) / sin(x-y) = (a+b) / (a-b)}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,}if a/b = c/d,

\textbf{then (a+b) / (a-b) = (c+d) / (c-d)}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}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}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}So the value of tanx/tany is a/b

\textbf{Hope It Helps}Hope It Helps