# Prove the identity using the t=tan 1/2 y results (tan y tan 1/2y) /( tan y - tan 1/2y) =sin y

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embizze | Certified Educator

`tan alpha/2 = +-sqrt((1-cos alpha)/(1+cos alpha))=(sin alpha)/(1+cos alpha)=(1-cos alpha)/(sin alpha)`

Choose the middle one:

`(tany tan(1/2y))/(tany - tan (1/2y))` convert tany to `siny/cosy`

`=(siny/cosy * siny/(1+cosy))/(siny/cosy - siny/(1+cosy))` Multiply the numerator and and the denominator

`=((sin^2y)/(cosy(1+cosy)))/((siny+sinycosy-sinycosy)/(cosy(1+cosy)))`

`=((sin^2y)/(cosy(1+cosy)))/(siny/(cosy(1+cosy)))`

`=(sin^2y)/siny`

`=siny` as required.