If `A=pi/5,` then find the value of `sum_(r=1)^8tan(r A)*tan((r+1)A)dot`

Explanation

`tan((r+1)A-(rA))=(tan(r+1)A-tan(rA))/(1+tan(r+1)A*tan(rA))`

`1+tan(rA)*tan(r+1)A=(tan(r+1)A-tan(rA))/(tanA)`

`tan(rA)*tan(r+1)A=(tan(r+1)A-tan(rA))/(tanA)-1`

`S=sum_(r=1)^8(tan(r+1)A-tan(rA))/(tanA)-sum_(r=1)^8 1`

`=1/(tanA)sum_(r=1)^8(tan(r+1)A-tan(rA)-8`

`=1/(tanA)[(tan2A-tanA)+(tan3A-tan2A)+...(tan9A-tan8A]-8`

`=1/(tanA)[tan9A-tanA]-8`

`tan9A=tan(9/5pi)=tan(2pi-pi/5)`

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