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Question

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

A
tan1(n2+nn2+n+2)
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B
tan1(n2+nn2n+2)
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C
tan1(n2+n+2n2+n)
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D
None of these
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Solution

The correct option is A tan1(n2+nn2+n+2)
We have,
nm=1tan1(2mm4+m2+2)
=nm=1tan1[2m1+(m2+m+1)(m2m+1)]
=nm=1tan1[(m2+m+1)(m2m+1)1+(m2+m+1)(m2m+1)]
=nm=1{tan1[m2+m+1]tan1[m2m+1]}
=tan13tan11+tan17tan13+(tan113tan17)+...+tan1(n2+n+1)tan1(n2n+1)
=tan1(n2+n+111+(n2+n+1)1)=tan1(n2+n2+n2+n)

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