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Question

(i) If tan A=56and tan B=111, prove that A+B=π4.
(ii) If tan A=mm-1and tan B=12m-1, then prove that A-B=π4.

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Solution

(i)
We have:tanA = 56 and tanB = 111Therefore, tanA+B = tanA+tanB1-tanA tanB tanA+B = tanA +tanB1 - tanA tanB tanA+B = 56+1111 - 56×111 tanA+B =61666166 tanA+B =1 tanA+B =tanπ4Therefore, A + B = π4.Hence proved.

(ii)
We know thattan(A-B)=tan A-tan B1+tan Atan B =mm-1-12m-11+m(m-1)(2m-1) =2m2-m-m+12m2-m-2m+1+m =2m2-2m+12m2-2m+1 =1A-B =tan-1(1) A-B =π4

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