interesting question

(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

suppose A+B=45°
=>tan(A+B)=1
=>(tanA+tanB)/(1-tanA.tanB)=1
=>tanA+tanB+tanAtanB=1
=>1+tanA+tanB+tanAtanB=1
=>(1+tanA)(1+tanB)=2

using this equation (1+tan1)(1+tan44)...tan45
=(2^22)*2
=2^23

thanks.....

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