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Question

nm=1tan1(2mm4+m2+2) is equal to

A
tan1(n2+n+1)π4
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B
tan1(n2+n+1)+π4
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C
tan1(n2+n1)π4
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D
tan1(n2n1)π4
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Solution

The correct option is A tan1(n2+n+1)π4
nm=1tan1(2mm4+m2+2) is equal to
In General formula for tan1x=tan1y is
tan1(x±y1±xy)...(1)
Now, given summation is,
nm=1tan1(2mm4+m2+2)=nm=1tan1(2m1+(m4+m2+1))
Now, factorisation of. m4+m2+1 is
m4+m2+1=(m2+m+1)×(m2m+1)
and difference between (m2m+1) and
(m2m+1) is 2m.
nm=1tan1((m2+m+1)(m2m+1)1+(m2+m+1)(m2m+1))...(2)
Now, compare eqn(1) & eqn(2)
=nm1tan1(m2+m+1)tan1(m2m+1)
Now putting value from 1, 2, ...,n.
=tan1(m2+1+1)tan1(121+1)+tan1(22+2+1)
tan1(222+1)+...+tan21(n2+n+1)..tan1(n2n+1)
=tan1(3)tan1(1)+tan1(7)tan1(3)+...+tan1(n2+n+1)tan1(n2n+1)
here alternative terms will canceled and we are left with tan1(n2+n+1)tan1(1)
So, tan1(n2+n+1)π4

1091872_1035489_ans_5d525eb1fd684fe390676dcf813605c7.jpg

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