(1+tan 1)(1+tan 2)(1+tan 3)(1+tan 4)(1+tan 5)......................(1+tan 45)= 2^n find the value of n answer:-23 how? prove by any theorem

Dear Aritra

Here you have to make pairs such that sum of angle is 45 like (1+tan1)(1+tan44) and (1+tan2)(1+tan43) etc there will be 22 pairs like this and one single term 1+tan45 which is equal to two

you have to apply sin(90-A)=cosA etc

you can write 1+tanA=√2 sin(45+A)/cosA=√2 cos(45-A)/cosA

like for pair (1+tan1)(1+tan44)=(√2 sin(45+1)/cos1)*(√2 cos(45-44)/cos44)

sin46 and cos44 will cancel the value of this pair will be 2. like this 22 pairs are there.it will be 2²² and last term is 1+tan45=2

so it will be 2²³

hence n=23

All the best.                                                           

AKASH GOYAL

AskiitiansExpert-IITD

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