Question

If $(1+\tan 5^{\circ })(1+\tan {10}^{\circ })(1+\tan {15}^{\circ })...(1+\tan {45}^{\circ })=2^k$, find the value of 'k' is

Answer 1

We have
$\tan ({45}^0-x)=\frac{1-\tan x}{1+\tan x}$
$(1+\tan ({45}^0-x\left)\right)-1=\frac{1-\tan x}{1+\tan x}$
$\Rightarrow (1+\tan ({45}^0-x\left)\right)(1+\tan x)=2$
Consider, $(1+\tan 5^{\circ })(1+\tan {10}^{\circ })(1+\tan {15}^{\circ })...(1+\tan {45}^{\circ })=2^k$
$\Rightarrow (1+\tan 5^{\circ })(1+\tan {40}^{\circ })(1+\tan {10}^{\circ })(1+\tan {35}^{\circ })(1+\tan {15}^{\circ })(1+\tan {30}^{\circ })(1+\tan {20}^{\circ })(1+\tan {25}^{\circ })(1+\tan {45}^{\circ })=2^k$
$\Rightarrow 2^5=2^k$
$\Rightarrow k=5$