Question

Given that $\left(1+\tan 1^{\circ }\right)\left(1+\tan 2^{\circ }\right)....\left(1+\tan {45}^{\circ }\right)=2^{\circ },$ find n.

Answer 1

Given that:$(1+\tan 1^{\circ })(1+\tan 2^{\circ })........(1+\tan {45}^{\circ })=2^n$

LHS,

$(1+\tan 1^{\circ })(1+\tan 2^{\circ })........(1+\tan {45}^{\circ })-\left(A\right)$

we know that

$\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A.\tan B},$ also, $\tan {45}^{\circ }=1$

So,

$\tan {45}^{\circ }=\frac{\tan 1^{\circ }+\tan {44}^{\circ }}{1-\tan 1^{\circ }.\tan {44}^{\circ }}\Rightarrow \tan 1^{\circ }+\tan {44}^{\circ }-(1-\tan 1^{\circ }.\tan {44}^{\circ })\left(\tan {45}^{\circ }\right)-\left(i\right)\tan {45}^{\circ }=\frac{\tan 2^{\circ }+\tan {43}^{\circ }}{1-\tan 2^{\circ }.\tan {43}^{\circ }}\Rightarrow \tan 2^{\circ }+\tan {43}^{\circ }-(1-\tan 2^{\circ }.\tan {43}^{\circ })\left(\tan {45}^{\circ }\right)-\left(ii\right)....\tan {45}^{\circ }=\frac{\tan {22}^{\circ }+\tan {23}^{\circ }}{1-\tan {22}^{\circ }.\tan {23}^{\circ }}\Rightarrow \tan {22}^{\circ }+\tan {23}^{\circ }-(1-\tan {22}^{\circ }.\tan {23}^{\circ })\left(\tan {45}^{\circ }\right)-\left(22\right)$

Using the above equation we can simplify the expression $\left(A\right)$

$(1+\tan 1^{\circ })(1+\tan 2^{\circ })(1+\tan 3^{\circ })........(1+\tan {44}^{\circ })(1+\tan {45}^{\circ })=(1+\tan 1^{\circ })(1+\tan 2^{\circ })(1+\tan 3^{\circ })........(1+\tan {44}^{\circ })\times 2=2(1+\tan 1^{\circ })(1+\tan 2^{\circ })(1+\tan 3^{\circ })........(1+\tan {44}^{\circ })=2(1+\tan 1^{\circ })(1+\tan {44}^{\circ })(1+\tan 2^{\circ })(1+\tan {43}^{\circ })(1+\tan 3^{\circ })(1+\tan {42}^{\circ }).....(1+\tan {22}^{\circ })(1+\tan {23}^{\circ })=2(\tan 1^{\circ }+\tan {44}^{\circ }+\tan 1.\tan 44+1)(\tan 2^{\circ }+\tan {43}^{\circ }+\tan 2^{\circ }.\tan {43}^{\circ }+1).......(\tan {22}^{\circ }+\tan {23}^{\circ }+\tan {22}^{\circ }.\tan {23}^{\circ }+1)=2\left[\right(\tan {45}^{\circ }\left)\right(1-\tan 1^{\circ }.\tan {44}^{\circ }+\tan 1^{\circ }.\tan {44}^{\circ }+1\left)\right(\tan {45}^{\circ }(1-\tan 2^{\circ }.\tan {43}^{\circ })+\tan 2^{\circ }.\tan {43}^{\circ }+1).......(\tan {45}^{\circ }(1-\tan {22}^{\circ }.\tan {23}^{\circ })+\tan {22}^{\circ }.\tan {23}^{\circ }+1\left)\right]=2\left[\right(1-\tan 1^{\circ }.\tan {44}^{\circ }+\tan 1^{\circ }.\tan {44}^{\circ }+1\left)\right(1-\tan 2^{\circ }.\tan {43}^{\circ }+\tan 2^{\circ }.\tan {43}^{\circ }+1).......(1-\tan {22}^{\circ }.\tan {23}^{\circ }+\tan {22}^{\circ }.\tan {23}^{\circ }+1\left)\right]=2.\left(2\right).\left(2\right)..........\left(2\right)=2.2^{22}=2^{23}\therefore n=23$