11suppose 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
The Product
(1 + tan1°) (1 + tan2°) (1 + tan3°) ........ (1 + tan45°)
equals:
a) 221
b) 222
c) 223
d) 224
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4 Answers
11(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
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