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)\...

* Hướng dẫn giải

Nhận xét: Nếu \(A+B={{45}^{0}}\) thì \(\left( 1+\tan A \right)\left( 1+\tan B \right)=2.\)

Thật vậy:

\(\left( 1+\tan A \right)\left( 1+\tan B \right)=\left( 1+\tan A \right)\left[ 1+\tan \left( {{45}^{0}}-A \right) \right]=\left( 1+\tan A \right)\left[ 1+\frac{\tan {{45}^{0}}-\tan A}{1+\tan {{45}^{0}}.\tan A} \right]\)

\(=\left( 1+\tan A \right)\left[ 1+\frac{1-\tan A}{1+\tan A} \right]=1+\tan A+1-\tan A=2.\)

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 \(P=a+b=22.\)