\(P=\left(tan1.tan89\right).\left(tan2.tan88\right)...\left(tan44.tan46\right).tan45\)
\(=\left(tan1.cot\left(90-89\right)\right).\left(tan2.cot\left(90-88\right)\right)...\left(tan44.cot\left(90-46\right)\right).1\)
\(=\left(tan1.cot1\right)\left(tan2.cot2\right)...\left(tan44.cot44\right)\)
\(=1.1...1=1\)