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Question

tan - _ + tan-I + tan- _ + tan

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Solution

We have to prove that tan 1 1 5 + tan 1 1 7 + tan 1 1 3 + tan 1 1 8 = π 4 .

Solve left hand side.

tan 1 1 5 + tan 1 1 7 + tan 1 1 3 + tan 1 1 8 = tan 1 ( 1 5 + 1 7 1 1 5 × 1 7 )+ tan 1 ( 1 3 + 1 8 1 1 3 × 1 8 ) = tan 1 ( 7+5 351 )+ tan 1 ( 8+3 241 ) = tan 1 ( 12 34 )+ tan 1 ( 11 23 ) = tan 1 ( 6 17 )+ tan 1 ( 11 23 )

Simplify further,

tan 1 1 5 + tan 1 1 7 + tan 1 1 3 + tan 1 1 8 = tan 1 ( 6 17 + 11 23 1 6 17 × 11 23 ) = tan 1 325 325 = tan 1 1 = π 4

It is equal to the right hand side.

Hence, it is proved that tan 1 1 5 + tan 1 1 7 + tan 1 1 3 + tan 1 1 8 = π 4 .


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