Is \( n \) Equal to 23 in the Given Trigonometric Product Equation?

[anemone]

Given that \(\displaystyle (1+\tan 1^{\circ})(1+\tan 2^{\circ})\cdots(1+\tan 45^{\circ})=2^n\), find $n$.

 

[kaliprasad]

Given that \(\displaystyle (1+\tan 1^{\circ})(1+\tan 2^{\circ})\cdots(1+\tan 45^{\circ})=2^n\), find $n$.

Note k in degrees and degree symbol not mentioned

tan (45-k)) = (tan 45- tank )/( 1+ tan 45 tan k)
= ( 1- tan k)/ (1 + tan k)
So 1+ tan (45-k) = ( 1+ tan k + 1 – tan k) / ( 1+ tan k) = 2/(1+ tan k)
Or (1 + tan (45-k))(1+ tan k) = 2

So (1 + tan 1) ( 1+ tan 44) = 2
(1+ tan 2)(1 + tan 43) = 2
( 1 + tan 22)(1+ tan 23) = 2

Hence (1+tan 1)( 1+ tan 2) … ( 1 + tan 44) = 2^22

As 1 + tan 45 = 2 so multiplying we get
(1+tan 1)( 1+ tan 2) … ( 1 + tan 44)( 1+ tan 45) = 2^23
hence n = 23