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Prove or disprove?

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cos(X+pi/6) + sin(x-pi/3) = 0

I am working with one side but I am not sure if I can subtract at all here.

  • Prove or disprove? - ,

    To see if it is an identity, pick any value of x
    e.g. x = 78°
    then x+π/6 ---> 78+30 = 108°
    and x-π/3 ---> 78 - 60 = 18°

    is cos108 + sin18 = 0 ???
    YES

    ok, then , let's prove it

    LS = cosxcosπ/6 - sinxsinπ/6 + sinxcosπ/3 - cosxsinπ/3
    = √3/2 cosx -1/2 sinx + 1/2 sinx - √3/2 cosx
    = 0
    = RS

  • Prove or disprove? - ,

    or, since we know that cos(pi/2 - x) = sin(x)

    cos(pi/2 - (x+pi/6)) = sin(pi/3 - x) = -sin(x-pi/3)

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