InterviewSolution
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InterviewSolution
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The value of `cot {sum_(23)^(n=1) cot^(-1)(1+ sum_(n=1)^(n) 2k)}` isA. `(23)/(25)`B. `(25)/(23)`C. `(23)/(24)`D. `( 24)/(23)` |
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Answer» Correct Answer - B We have , ` cot [ overset ( 23) underset ( n = 1 ) ( sum) cot ^(-1) (1 + overset n underset (h = 1 ) (sum) 2k )]` `rArr cot [ overset ( 23) underset(n = 1 ) (sum) cot ^(-1) (1 + 2 + 4 + 6 + 8 + ... + 2n )]` `rArr cot [ overset ( 23) underset ( n = 1) ( sum) cot ^(-1) { 1 + n (n + 1 )}] ` ` rArr cot [ overset ( 23) underset ( n = 1 ) (sum) tan ^(-1) ""(1)/( 1+ n (n + 1 ))]` `rArr cot [ overset ( 23) underset ( n = 1) ( sum) tan ^(-1) { (( n + 1 ) - n )/( 1+ n ( n + 2 ))}]` `rArr cot [ overset ( 23) underset ( n = 1 ) ( sum) ( tan ^(-1) ( n + 1 ) - tan ^(-1) In n ) ]` `rArr cot [ (tan ^(-1) 2 - tan ^(-1) 1 ) + ( tan ^(-1) 3 - tan ^(-1) 2) + ( tan ^(-1) 4 - tan ^(-1 ) 3) ] + ... + ( tan ^(-1) 24 - tan ^(-1) 23)]` `rArr cot ( tan ^(-1) 24 - tan ^(-1) 1)` `rArr cot ( tan ^(-1) ""( 24 - 1 ) /( 1+ 24 *(1))) = cot ( tan ^(-1) ""( 23)/( 25))` `" " = cot ( cot ^(-1) ""( 25)/( 23)) = ( 25)/( 23)` |
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