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If A=pi/5, then find the value of sum(r=...

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

Text Solution

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`tan((r+1)A-(rA))=(tan(r+1)A-tan(rA))/(1+tan(r+a)A.tan(rA))`
`rArr S=sum_(r=1)^(8)tan(rA)tan(r+1)A`
`=sum_(r=1)^(8)(-1)+(1)/(tanA)sum_(r=1)^(8)(tan(r+1)A-tan(rA))`
`=-8+(1)/(tan A)(tan 9A-tanA)`
Now `tan 9A=tan""(9pi)/(5)=tan(2pi-(pi)/(5))=-tan""(pi)/(5)`
`rArr S=-8+(1)/(tan A)(-2tanA)=-8-2=-10`
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