Complex Trigo #1

There is an identity that,

(1+tanA)(1+tanB) = 2 if A+B = 45 degrees.

The proof is actually very simple. Just expand.

We get 1-tanAtanB = TanA+TanB. which also means that tan(A+B) = tan(45 degrees) = 1.

So, as there are 22 pairs in the question, the answer is 2^22.

${when~~x+y=45º}$ $\frac{\tan(x)+\tan(y)}{1-\tan(x)\times\tan(y)}=1$ $\therefore \tan(x)+\tan(y)=1-\tan(x)\times\tan(y)$ $\tan(x)+\tan(y)+\tan(x)\times\tan(y)~~~~\color{#3D99F6}{(+1)}=1~~~~\color{#3D99F6}{(+1)}$ $\tan(x)+\tan(y)+\tan(x)\times\tan(y)+1=2={\boxed{(1+\tan(x))(1+\tan(y))}}$ $P={(1+\tan 1^\circ)(1+\tan 2^\circ)(1+\tan 3^\circ)\cdots(1+\tan 43^\circ)(1+\tan 44^\circ)}$ $P=(1+\tan(1^\circ))(1+\tan(44^\circ))(1+\tan(2^\circ)(1+\tan(43^\circ))...$ $P=\underbrace { 2\times 2\times \dots \times 2 }_{ 22 \, \text{times} } = {2}^{22}$ $2^{m}=2^{22}$ $\color{#3D99F6}{\boxed{m=22}}$

(1+tan44) = 2/(1+tan1).. similarly (1+tan 43) = 2/(1+tan42) we do this till tan23 .. multiplying will cancel out all and remainder will be 2^22 as there are 22 numbers till 23 from 45