mathIV

HOW TO PROVE TANGENT THETA +(SECANT THETA--1 DIVIDE TANGENT THETA-SECANT THETA+1=TANGENT THETA+SECANT THETA

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  1. tanØ + secØ - 1 / tanØ - secØ + 1 = tanØ + secØ

    Surely , you must have meant

    (tanØ + secØ - 1) / (tanØ - secØ + 1) = tanØ + secØ

    I will use x instead of Ø for easier typing

    LS = (sinx/cosx + 1/cosx - cosx/cosx) / (sinx/cosx - 1/cosx + cosx/cosx)
    = (sinx + 1 - cosx)/(sinx - 1 + cosx)
    = (sinx + 1 - cosx)/(sinx - 1 + cosx) * (sinx - 1 - cosx)/(sinx - 1 - cosx)
    carefully expanding this, and replacing 1 with sin^2 x + cos^2 x , I got
    = -2sinx cosx/(2sin^2 x - 2sinx)
    = cosx/(1 - sinx)

    RS = sinx/cosx + 1/cosx
    = (sinx + 1)/cosx
    = (sinx + 1)/cosx * (sinx - 1)/(sinx - 1)
    = (sin^2 x - 1)/(sinxcosx - cosx)
    = -cos^2 x/(cosx(sinx - 1))
    = cosx/(1 - sinx)
    = LS

    Whewww!!!!
    There has to be an easier way.

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    posted by Reiny

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