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trig

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Given that tan 45=1, use tan(x+y) to show that tan 22.5= /2 - 1.
(/is square root sign

  • trig - ,

    tan(x+y)=(tan(x)+tan(y))/(1-tan(x)tan(y))

    substitute x=y to get
    tan(2x) = 2tan(x)/(1-tan²(x))
    Substitute tan(2x)=1 and solve for tan(x).
    Reject the negative root knowing that tan(x)≥0 for 0<x<π/2.

  • trig - ,

    tan(x+y) = (tanx + tany)/(1 - tanxtany)

    so tan 45
    = tan(22.5 + tan22.5)/(1 - tan22.5tan22.5)
    = 2tan22.5)/1 - tan^2 22.5)
    let tan22.5 = y for ease of typing

    1 = 2y/(1 - y^2)
    1 - y^2 = 2y
    y^2 + 2y - 1 = 0
    y^2 + 2y = 1 , by completing the square
    y^2 + 2y + 1 = 1+1
    (y+1)^2 = 2
    y+1 = √2
    y = √2-1

    tan22.5° = √2 - 1

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