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# (1+TAN1)(1+TAN2).....................(1+TAN45) =? Priyansh Bajaj AskiitiansExpert-IITD
30 Points
10 years ago

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].

Please feel free to post as many doubts on our discussion forum as you can. If you find any question
Difficult to understand - post it here and we will get you the answer and detailed solution very quickly. We are all IITians and here to help you in your IIT JEE preparation.

All the best Akash!!!

Regards,
Priyansh Bajaj Kushagra Madhukar
9 months ago
Dear student,

[(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

Thanks and regards,
Kushagra