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

Explanation

We have,
`tan((pi)/(4)+(y)/(2))=tan^(3)((pi)/(4)+(x)/(2))`
`rArr(tan pi//4+tany//2)/(1-tanpi//4tany//2)=[(tan pi//4+tanx//2)/(1-tan pi//4tanx//2)]^(3)`
`rArr(1+tany//2)/(1-tany//2)=((1+tanx//2)/(1-tanxx/2))^(3)`
`rArr (cos y//2+siny//2)/(cos y//2-siny//2)=((cosx//2+sinx//2)/(cosxx/2-sinx//2))^(3)`
`rArr((cos y//2+siny//2)/(cos y//2-siny//2))^(2)=[((cosx//2+sinx//2)/(cos x//2-sinx//2))^(2)]^(3)`
`rArr(1+2siny//2cosy//2)/(1-2siny//2cosy//2)=((1+2sinx//2cosx//2)/(1-2sinx//2cosx//2))^(3)`
`rArr(1+siny)/(1-siny)=((1+sinx)/(1-sinx))^(3)`
Applying componendo and dividendo
`rArr ((1+siny)-(1-siny))/((1+siny)+(1-siny))=((1+sinx)^(3)-(1-sinx)^(3))/((1+sinx)^(3)+(1-sinx)^(3))`
`(2siny)/(2)=(6sinx+2sin^(3)x)/(2+6sin^(2)x)`
`rArr (sin y)/(sinx)=(3+sin^(2)x)/(1+3sin^(2)x)`