Tangential problem

Geometry Level 3

( 1 + tan ⁡ 1 ∘ ) ( 1 + tan ⁡ 2 ∘ ) ( 1 + tan ⁡ 3 ∘ ) ⋯ ( 1 + tan ⁡ 4 5 ∘ ) \large (1+\tan 1^{\circ})(1+\tan 2^{\circ})(1+\tan 3^{\circ}) \cdots (1+\tan 45^{\circ})

If the value of the expression above equal to 2 n 2^n , find the value of n n .


The answer is 23.

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Subhaditya Basu by Subhaditya Basu , 20, India

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2 solutions

Maria Catalin
Jul 21, 2015

we use the formula tan(a)*tan(b)=1-(tan(a)+tan(b))/tan(a+b)

(1+tan1)(1+tan44)=1+tan44+tan1+tan1*tan44=1+tan44+tan1+1-(tan1+tan44)/tan45=2

using same method (1+tan2)(1+tan43)=2 etc

in conclusion the result is 2^23 therefore n=23

4 Helpful 0 Interesting 0 Brilliant 0 Confused
Aakash Khandelwal
Jul 21, 2015

Use (1+tanx)(1+tan(45-x))=2

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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