The product (1 + tan1°) (1 + tan2°) (1 + tan3°) ………. (1 + tan 45°) equal to ?

We can start with :
=(1+tan 0^o)(1 + tan1°) (1 + tan2°) (1 + tan3°) ………. (1 + tan 45°) {Since , tan 0^o = 0}

Now , we know that
tan(A + B) = [ tan(A) + tan(B) ] / [ 1 - tan(A) tan(B) ]
And tan(45) = 1

So
tan(45) = [ tan1 + tan44 ] / [ 1 - tan1 tan44 ]
tan(45) = [ tan2 + tan43 ] / [ 1 - tan2 tan43 ]
.
.
.
tan(45) = [ tan22 + tan23 ] / [ 1 - tan22 tan23 ]

So we have ,
tan1 + tan44 = tan(45) [ 1 - tan1 tan44 ]
tan2 + tan43 = tan(45) [ 1 - tan2 tan43 ]
.
.
.
tan22 + tan23 = tan(45) [ 1 - tan22 tan23 ]




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So finally we have ,

(1 + tan1) (1 + tan2) (1 + tan3) ... (1 + tan44) (1 + tan45)

= (1 + tan1) (1 + tan2) (1 + tan3) ... (1 + tan44) (1 + 1)

= 2 (1 + tan1) (1 + tan2) (1 + tan3) ... (1 + tan44)

= 2 (1 + tan1) (1 + tan44) * (1 + tan2) (1 + tan43) * (1 + tan3) (1 + tan42) * ... * (1 + tan22) (1 + tan23)

= 2 (tan1 + tan44 + tan1 tan44 + 1) (tan2 + tan43 + tan2 tan43 + 1) ... (tan22 + tan23 + tan22 tan23 + 1)

= 2 ((tan45)(1 - tan1 tan44) + tan1 tan44 + 1) ((tan45)(1 - tan2 tan43) + tan2 tan43 + 1) ... ((tan45)(1 - tan22 tan23) + tan22 tan23 + 1)

= 2 (1 - tan1 tan44 + tan1 tan44 + 1) (1 - tan2 tan43 + tan2 tan43 + 1) ... (1 - tan22 tan23 + tan22 tan23 + 1)

= 2 (2) (2) ... (2)

= 2 (2^22)

= 2^23

= 8388608

tan45° = (tana + tanb)/(1 - tana tanb) ........... where a + b = 45 { compound angle formula }
tana + tanb = (1 - tana tanb)tan45°
tana + tanb = 1 - tana tanb



(1 + tan 1°) (1 + tan 2°) (1 + tan 3°) ··· (1 + tan 45°)

= (1 + tan 1°)(1 + tan 44°) * (1 + tan 2°)(1 + tan 43°) * (1 + tan 3°)(1 + tan 42°) * .... (1 + tan 22°)(1 + tan 23°) * (1 + tan 45°)

= (1 + tan 1°)(1 + tan 44°) * (1 + tan 2°)(1 + tan 43°) * (1 + tan 3°)(1 + tan 42°) * .... (1 + tan 22°)(1 + tan 23°) * (1 + 1)

= (1 + tan 1°)(1 + tan 44°) * (1 + tan 2°)(1 + tan 43°) * (1 + tan 3°)(1 + tan 42°) * .... (1 + tan 22°)(1 + tan 23°) * 2

= 2(1 + tan 1°)(1 + tan 44°) * (1 + tan 2°)(1 + tan 43°) * (1 + tan 3°)(1 + tan 42°) * .... (1 + tan 22°)(1 + tan 23°)

= 2(1 + tan44° + tan1° + tan1° tan44°) * (1 + tan43° + tan2° + tan43° tan2°) * (1 + tan43° + tan3° + tan42° tan3°) * ... * (1 + tan22° + tan23° + tan22° tan23°)

= 2(1 + 1 - tan1° tan44° + tan1° tan44°) * (1 + 1 - tan2° tan43° + tan2° tan43°) * (1 + 1 - tan3° tan43° + tan3° tan43°) * ... * (1 + 1 - tan22° tan23° + tan22° tan23°)

= 2(1 + 1)(1 + 1)(1 + 1) ... (1 + 1) .............................. a total of 23 factors of (1 + 1)

= 2(1 + 1)²²

= 2(2)²²

= 2²³ = 2ⁿ

n = 23