Question

Could you demonstrate this equation?! And thanks you btw! 3arctan(2-sqr(3))=arctan1/2 + arctan1/3

Answer 1

Use, `(2-sqrt3)=tan15^0` to prove the equation.

Explanation:

Here, `3tan^(-1)(2-sqrt3)=tan^-1(1/2)+tan^-1(1/3)`
We know that,`color(red)(tan15^0)=tan(60^0-45^0)`
`=>tan15^0=(tan60^0-tan45^0)/(1+tan60^0tan45^0`
`=>tan15^0=(sqrt(3)-1)/(1+sqrt3)=((sqrt(3)-1)(sqrt(3)-1))/((sqrt(3)+1)(sqrt(3)-1))=(3-2sqrt(3)+1)/(3-1)=(4-2sqrt3)/2=color(red)((2-sqrt3)`
`LHS=3tan^(-1)(2-sqrt3)=3tan^1(tan15^0)=3*15^0=45^0`
`color(blue)(=>LHS=pi/4)`
`RHS=tan^-1((1/2+1/3)/(1-1/2*1/3))`
`=tan^-1(((3+2)/6)/((6-1)/6))`
`=tan^-1(5/5)=tan^-1(1)`
`color(blue)(RHS=pi/4)`