(1+TAN1)(1+TAN2).....................(1+TAN45) =?

Dear Akash,

Solution:-  Club the terms like this-

[(1+tan1)(1+tan44)] * [(1+tan2)(1+tan43)] * [(1+tan3)(1+tan42)].......[(1+tan22)(1+tan23)]*[(1+tan45)]

= [(1+(tan1+tan44) + (tan1)(tan44)] * [(1+(tan2+tan43) + (tan2)(tan43)]......[(1+(tan22+tan23) + (tan22)(tan23)]* 2

Since, tanA + tan B = tan(A+B) [1- (tanA)(tanB)]. So, (tan1 +tan44) = tan(45) [1- (tan1)(tan44)] = [1- (tan1)(tan44)]. Doing it for all terms and putting them in above expression will give-

[(1+(1- (tan1)(tan44)) + (tan1)(tan44)] * [(1+(1- (tan2)(tan43)) + (tan2)(tan43)]......[(1+(1- (tan22)(tan23)) + (tan22)(tan23)]* 2

= (2)*(2)*(2)......(2)*(2) = 2²³ [ANS].                        

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All the best Akash!!!

Regards,
Askiitians Experts
Priyansh Bajaj