If `tan(pi/4+y/2)=tan^3(pi/4+x/2)dot` Prove that `(siny)= (sinx)(3+sin^2x)/(1+3sin^2x)` .

Explanation

`(tanpi/4+tany/2)/(1-tanpi/4*tany/2)=[(tan(pi/4)+tan(x/2))/(1-tan(pi/4)*tan(x/2))]^3`

`(1+tan(y/2))/(1-tan(y/2))=[(1+tan(x/2))/(1-tan(x/2))]^3`

`(cos(y/2)+sin(y/2))/(cos(y/2)-sin(y/2))=(((cos(x/2)+sin(x/2))/(cos(x/2)-sin(x/2)))^2)^3`

`(cos^2(y/2)+sin^2(y/2)+2cos(y/2)*sin(y/2))/(cos^2(y/2)+sin^2(y/2)-2cos(y/2)*sin(y/2))=[(cos^2(x/2)+sin^2(x/2)+2cos(x/2)sin(y/2))/(cos^2(x/2)+sin^2(x/2)-2cos(x/2)sin(y/2))]^3`

`(1+siny)/(1-siny)=((1+sinx)/(1-sinx))^3`

`(1+siny+1-siny)/(1+siny-1+siny)=((1+sinx)^3+(1-sinx)^3)/((1+sinx)^3+(1-sinx)^3)`

`2/(2siny)=(2(1+sinx)^2-(1+sinx)(1-sinx)+(1-sinx)^2)/(2sinx((1+sinx)^2+(1+sinx)(1-sinx)+(1-sinx)^2)`

`1/siny=1/sinx((3sin^2x+1)/(3+sin^2x))`

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