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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