trigonometry in handy...

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 ⇦ ANSWER

1+ tan1 = 1+ tan (45 - 44) = $\frac{2}{1+ tan 44}$ similarly do for others

let 45=a+b tan 45 = (tan a + tan b)/(1 - tan a tan b) 1 - tan a tan b = tan a + tan b (1 + tan a)(1 + tan b)= 2 Thus the required product = 2^23