Prove that: `t a nthetatan(60^0-theta)tan(60^0+theta)=tan3thetadot`

Explanation

We know that, `tan (a-b)=frac{tan a-tan b}{1+tan a tan b}` and `tan(a+b)=frac{tan a+tan b}{1-tan a tan b}`

`tan theta[frac{tan 60^@-tan theta}{1+tan 60^@ tan theta}][frac{tan 60^@+tan theta}{1-tan 60^@ tan theta}]`

`=tan theta[frac{sqrt(3)-tan theta}{1+sqrt(3) tan theta}][frac{sqrt(3)+tan theta}{1-sqrt(3) tan theta}]`

`=tan theta(frac{3-tan^2 theta}{1-3tan^2 theta})`

`=frac{3tan theta-tan^3 theta}{1-3tan^2 theta}`

`=tan 3 theta`