Prove that (1+tan1°) (1+tan2°).........(1+tan45°)=2 to the power 23

Dear Deepak

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

Regards
Arun (askIITians forum expert)

Approved