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Find the value of sum (r=1)^(89) "log"(1...

Find the value of `sum _(r=1)^(89) "log"_(10) "cot" (pir)/(180)`

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We have ,
`sum _(r=1)^(89) "log"_(10) "cot" (pir)/(180)`
` "log"_(10) "cot" (pi)/(180)+log_(10) "cot"(2pi)/(180)+ log_(10)"cot"(3pi)/(180) + ...+ (88pi)/(180) + log_(10) "cot" (89pi)/(180)`
` log_(10) cot 1^(@)+log_(10) cot 2^(@)+log_(10) cot 3^(@)+...+log_(10) cot 88^(@)+log_(10) cot 89^(@)[ because (pi^(c))/(180) = 1 ^(@)]`
`log_(10) (cot 1^(@) cot 2^(@) cot 3^(@)....... cot 88^(@) cot 89^(@))`
`log_(10) {(cot 1^(@) cot 89^(@))( cot 2^(@) cot 88^(@))(cot3^(@) cot 87^(@))...(cot44^(@) cot 45^(@)) cot45^(@)}`
`log_(10) 1 = 0 `
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