# Pre-calculus

prove sin(x+y)+sin(x-y)/cos(x+y)+cos(x-y)=tanx

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1. Use trigonometric identities:

sin ( x + y ) = sin x cos y + cos x sin y

sin ( x - y ) = sin x cos y - cos x sin y

cos ( x + y ) = cos x cos y - sin x sin y

cos ( x - y ) = sin x sin y + cos x cos y

sin ( x + y ) + sin ( x - y ) =

sin x cos y + cos x sin y + sin x cos y - cos x sin y =

sin x cos y + sin x cos y + cos x sin y - cos x sin y =

2 sin x cos y

cos ( x + y ) + cos ( x - y ) =

cos x cos y - sin x sin y + sin x sin y + cos x cos y =

cos x cos y + cos x cos y - sin x sin y + sin x sin y =

2 cos x cos y

[ sin ( x + y ) + sin ( x - y ) ] / [ cos ( x + y ) + cos ( x - y ) ] =

2 sin x cos y / 2 cos x cos y =

sin x / cos x = tan x

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2. Watch those brackets, you must have meant:
(sin(x+y)+sin(x-y))/(cos(x+y)+cos(x-y))=tanx

LS =
(sinxcosy + cosxsiny + sinxcosy - sinxcosy)/(cosxcosy - sinxsiny + cosxcosy + sinxsiny)
= 2sinxcosy/(2cosxcosy)
= (sinx/cosx)(cosy/cosy)
= tanx
= RS

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2. 👎 1

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