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Question

If tan1(x1x2)+tan1(x+1x+2)=n4. Find x

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Solution

If tan1x1x2+tan1x+1x+2=π4

We know than
tan1x+tan1y=tan1(x+y)(1xy)

tan1(x1x2)+tan1x+1x+2=tan1⎢ ⎢ ⎢x1x2+x+1x+21x1x2.x+1x+2⎥ ⎥ ⎥

=tan1⎢ ⎢ ⎢ ⎢ ⎢(x1)(x+2)+(x+1)(x2)(x2)(x+2)(x2)(x+2)(x1)(x+2)(x2)(x+2)⎥ ⎥ ⎥ ⎥ ⎥

=tan1[(x1)(x+2)+(x+1)(x2)(x2)(x+2)(x1)(x+2)]

=tan1[x2+2xx2+x22x+x2x2x24+1]

=tan1[2x2y3]

tan1[2x2y3]=π4

2x2y3=tanπ4

2x243=1

2x24=3

2x2=43=1

x2=12

x=±12

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