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Câu hỏi: Gọi $a,b$ là các số nguyên thỏa mãn $\left( 1+\tan {{1}^{o}} \right)\left( 1+\tan {{2}^{o}} \right)...\left( 1+\tan {{43}^{o}} \right)={{2}^{a}}.\left( 1+\tan {{b}^{o}} \right)$ đồng thời $a,b\in \left[ 0;90 \right].$ Tính $P=a+b.$
A. 46.
B. 22.
C. 44.
D. 27.
A. 46.
B. 22.
C. 44.
D. 27.
Nhận xét: Nếu $A+B={{45}^{0}}$ thì $\left( 1+\tan A \right)\left( 1+\tan B \right)=2.$
Thật vậy:
$\left( 1+\tan A \right)\left( 1+\tan B \right)=\left( 1+\tan A \right)\left[ 1+\tan \left( {{45}^{0}}-A \right) \right]=\left( 1+\tan A \right)\left[ 1+\dfrac{\tan {{45}^{0}}-\tan A}{1+\tan {{45}^{0}}.\tan A} \right]$
$=\left( 1+\tan A \right)\left[ 1+\dfrac{1-\tan A}{1+\tan A} \right]=1+\tan A+1-\tan A=2.$
Khi đó:
$\left( 1+\tan {{1}^{0}} \right)\left( 1+\tan {{2}^{0}} \right)\left( 1+\tan {{3}^{0}} \right)...\left( 1+\tan {{42}^{0}} \right)\left( 1+\tan {{43}^{0}} \right)=$
$=\left( 1+\tan {{1}^{0}} \right)\left[ \left( 1+\tan {{2}^{0}} \right)\left( 1+\tan {{43}^{0}} \right) \right]\left[ \left( 1+\tan {{3}^{0}} \right)\left( 1+\tan {{42}^{0}} \right) \right]...\left[ \left( 1+\tan {{22}^{0}} \right)+\left( 1+\tan {{23}^{0}} \right) \right]$
$=\left( 1+\tan {{1}^{0}} \right){{.2}^{21}}$. Suy ra $a=21,b=1.$
Vậy $P=a+b=22.$
Thật vậy:
$\left( 1+\tan A \right)\left( 1+\tan B \right)=\left( 1+\tan A \right)\left[ 1+\tan \left( {{45}^{0}}-A \right) \right]=\left( 1+\tan A \right)\left[ 1+\dfrac{\tan {{45}^{0}}-\tan A}{1+\tan {{45}^{0}}.\tan A} \right]$
$=\left( 1+\tan A \right)\left[ 1+\dfrac{1-\tan A}{1+\tan A} \right]=1+\tan A+1-\tan A=2.$
Khi đó:
$\left( 1+\tan {{1}^{0}} \right)\left( 1+\tan {{2}^{0}} \right)\left( 1+\tan {{3}^{0}} \right)...\left( 1+\tan {{42}^{0}} \right)\left( 1+\tan {{43}^{0}} \right)=$
$=\left( 1+\tan {{1}^{0}} \right)\left[ \left( 1+\tan {{2}^{0}} \right)\left( 1+\tan {{43}^{0}} \right) \right]\left[ \left( 1+\tan {{3}^{0}} \right)\left( 1+\tan {{42}^{0}} \right) \right]...\left[ \left( 1+\tan {{22}^{0}} \right)+\left( 1+\tan {{23}^{0}} \right) \right]$
$=\left( 1+\tan {{1}^{0}} \right){{.2}^{21}}$. Suy ra $a=21,b=1.$
Vậy $P=a+b=22.$
Đáp án B.