maths (trigonometry)

1+sin a- cos a/1+sin a +cos a eual to ?.
How?

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  1. Multiply both numerator and denominator by the conjugate, (1+sin(a)-cos(a))

    To reduce typing, will use s=sin(a), c=cos(a),
    so
    (1+sin(a)- cos(a))/(1+sin(a)+cos(a))
    =(1+s-c)/(1+s+c)
    =(1+s-c)²/((1+s+c)(1+s-c))
    =(1+s²+2s+c²-2c(1+s))/((1+s)²-c²)
    now use s²+c²=1
    =((2+2s)-2c(1+s))/(1+2s+s²-c²)
    =2(1-c)(1+s)/(2s(1+s))
    cancel 2(1+s) assume s≠-1
    =(1-c)/s
    =(1-cos(a))/sin(a)

    Check my work.

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