(1+tan 1)(1+tan 2)(1+tan 3).....(1+tan 44) = 2^k

What is the value of "k" given that all the angles are in degrees ?

 (1+tan1)(1+tan2)...................(1+tan44)

=(1+sin1/cos1)(1+sin2/cos2)....................(1+sin44/cos44)

=[(sin1+cos1)(sin2+cos2)(sin3+cos3).......................(sin44+cos44)]/(cos1cos2cos3...cos44)

=sqrt(2)sin(45+1)sqrt(2)sin(45+2)sqrt(2)sin(45+3)........................sqrt(2)sin(45+44)/(cos1cos2cos3...cos44)

apply sin @ = cos (90 - @)

then u will be left with

sqrt(2)sqrt(2)sqrt(2)sqrt(2)sqrt(2)..........44 times

ie 2^(44/2) = 2^22

hence proceed

k = 22

1 + tan(44) = 1 + tan(45 - 1) = 1 + [ 1 - tan(1) / 1 + tan(1) ] = 2 / ( 1 + tan(1) )

do same for terms from end. See from above that denominator cancels out with ( 1 + tan1 ) in numerator. Like this all the terms cancel out giving the product equal to 2²². So, k = 22.

Considering all angles to be in degrees

 If  A + B = 45 

 then (1 + tan A)(1+tan B) = 2

Proof :- 

A + B = 45 

if  A = 1  and B = 44

(1+tan 1) ( 1+tan 44) = 1 + tan 44 + tan 1 + tan 1. tan 44    ...(1)

tan (45 ) = tan (44+ 1) = (tan 1 + tan 44) /  (1 - tan1. tan 44 )

The value of  tan 45 = 1

1 =( tan 1 + tan 44 ) / (1 - tan 1 . tan 44)

or 1 - tan 1.tan44 = tan 1 + tan44 

or 1 = tan 1 + tan 44 + tan 1. tan 44   

Thus ( 1+tan 1)(1+tan 44) = 1+1 = 2

(1 + tan 45) = 2

(1+ tan 1)(1+tan 44) = 2

(1 + tan 2) (1+tan 43) = 2

(1+tan 3)(1+tan 42) = 2

(1+tan 4)(1+ tan 41) = 2

(1+tan 5) (1+tan 40) = 2

(1+tan 6)(1+tan 39) = 2

(1+tan 7)(1+tan 38) = 2

(1+tan 8)(1+tan 37)=2

(1+tan 9)(1+tan 36)=2

(1+tan 22)(1+tan 23)= 2

so all are just like above      

2^k = 2^22