If `("tan"(theta+alpha))/a=""("tan"(theta+beta))/b=""("tan"(theta+gamma))/c` then prove `(a+b)/(a-b)sin^2(alpha-beta)+(b+c)/(b-c)sin^2(beta-gamma)+(c+a)/(c-a)sin^2(gamma-alpha)=0`

Explanation

`x/y=(tan(theta+alpha))/(tan(theta+beta))`

`(x+y)/(x-y)=(tan(theta+alpha)+tan(theta+beta))/(tan(theta+alpha)-tan(theta+beta))`

`(x+y)/(x-y)=(sin(2theta+alpha+beta))/(sin(alpha-beta))`

`(x+y)/(x-y)sin^2(alpha-beta)=sin(2theta+alpha+beta)sin(alpha-beta)-(1)`

`=1/2(cos2(theta+beta)-cos2(theta+alpha))`

`(y+z)/(y-z)sin^2(beta-gamma)=1/2[cos2(theta+gamma)-cos2(theta+beta))-(2)`

`(z+x)/(z-x)sin^2(gamma-alpha)=1/2[cos2(theta+alpha)-cos2(theta+gamma)]-(3)`

adding equation 1,2and3

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