# A “fast” way for computing $\prod \limits_{i=1}^{45}(1+\tan i^\circ)$?

Which is the fastest paper-pencil approach to compute the product $$\prod \limits_{i=1}^{45}(1+\tan i^\circ)$$

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Using $$1+\tan x = \frac{\sin x + \cos x}{\cos x} = \frac{\sqrt{2} \cos (45^{\circ} - x)}{\cos x},$$ the product can be written as: $$\prod_{x=1}^{45}(1+\tan x^\circ) = 2^{45/2} \prod_{x=1}^{45} \frac{\cos (45 - x)^{\circ}}{\cos x^{\circ}} \stackrel{(1)}{=} 2^{45/2} \cdot \frac{\prod\limits_{x=0}^{44} \cos x^{\circ}}{\prod\limits_{x=1}^{45} \cos x^{\circ}} \stackrel{(2)}{=} 2^{45/2} \cdot \frac{\cos 0}{\cos 45^{\circ}} = 2^{23},$$ where we

1. reindexed the product in the numerator, and
2. cancelled the common factors.

Another approach. If $x+y = 45^{\circ}$, then $$1 = \tan(x+y) = \frac{\tan x + \tan y}{1 - \tan x \tan y},$$ which rearranges to $$\tan x \tan y + \tan x + \tan y = 1 \quad \implies \quad (1+\tan x)(1+\tan y) = 2.$$ Now plug in $x = 0^{\circ}, 1^{\circ}, 2^{\circ}, \ldots, 45^{\circ}$, so that $y$ takes the same values but in the opposite order. Multiplying all these equations, we get $$\left[ \prod_{x=0}^{45} (1+\tan x^\circ) \right]^2 = 2^{46}.$$ Taking square-roots and noting that $1+\tan 0^\circ = 1$, we get the answer.

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I like the second answer even better. I wish I could vote up again! – JavaMan Oct 25 '11 at 20:39
+1. Quite nice. – Did Oct 25 '11 at 20:42
@DJC:Second answer is nice,but I guess not much of intuitive under exam constraints if you haven't did something very similar to this before. – Quixotic Oct 25 '11 at 21:24
Amazing idea - Very neat! – NoChance Oct 26 '11 at 7:46

Using this, $$(\cot A + \tan y)(\cot A+ \tan(A-y))=\csc^2A \text{ if } A\ne m\pi\text{ where }m\text{ is any integer}$$

Putting $A=45^\circ, (1 + \tan y)(1+ \tan(45^\circ-y))=\csc^245^\circ=2$

Now, putting $y=1^\circ,2^\circ,3^\circ,\cdots,\lfloor\frac{45}2\rfloor^\circ=22^\circ$ and multiplying them we get, $$\prod_{1\le r\le 22}(1+\tan r^\circ)(1+\tan(45-r)^\circ)=2^{22}$$

$$\implies \prod_{1\le r\le 44}(1+\tan r^\circ)=2^{22}$$

The unpaired $1+\tan45^\circ=1+1=2$

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Just tell a computer to calculate them. For example in R this runs almost instantly

> prod(1+tan((1:45)*pi/180))
[1] 8388608

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This is actually quantitative aptitude question,which requires pencil-paper approach only. – Quixotic Oct 25 '11 at 19:51
Perhaps you should have put that in the question at the start – Henry Oct 25 '11 at 21:03
@Henry: It was kind of obvious. – TonyK Oct 25 '11 at 21:09
No matter what others say, this is the fast way – Norbert Jan 2 '12 at 17:34