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Question

m=1ntan-12mm4+m2+2 is equal to


A

tan-1n2+nn2+n+2

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B

tan-1n2-nn2-n+2

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C

tan-1n2+n+2n2+n

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D

None of these

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Solution

The correct option is A

tan-1n2+nn2+n+2


Explanation for the correct option :

Step-1 : (Simplify the expression m4+m2+2)

m4+m2+2=m22+2m2+1-m2+1=m2+12-m2+1=1+1+m+m21-m+m2

Step-2 : Simplify the expression 2mm4+m2+2

We can write, 2m=1+m+m2-1-m+m2

So, 2mm4+m2+2=1+m+m2-1-m+m21+1+m+m21-m+m2

Step-3 : Evaluate tan-12mm4+m2+2

Formula to be used : We know that tan-1x-tan-1(y)=tan-1x-y1+xy

So, we get the following identity

tan-12mm4+m2+2=tan-11+m+m2-1-m+m21+1+m+m21-m+m2=tan-11+m+m2-tan-11-m+m2

Step-4 : Evaluate m=1ntan-12mm4+m2+2

Using the identity obtained in Step-3, we get

m=1ntan-12mm4+m2+2=m=1ntan-11+m+m2-tan-11-m+m2=tan-11+1+12-tan-11-1+12+tan-11+2+22-tan-11-2+22+tan-11+3+32-tan-11-3+32++tan-11+n+n2-tan-11-n+n2=tan-13-tan-11+tan-17-tan-13+tan-113-tan-17++tan-11+n+n2-tan-11-n+n2=tan-11+n+n2-tan-11=tan-11+n+n2-11+1+n+n2.1tan-1(x)-tan-1(y)=tan-1x-y1+xy=tan-1n2+nn2+n+2

Hence, option (A) is the correct answer.


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