Fun Tan Question

Question

Using only trig identities, how would you approach the following question?

Determine the value of $$ \prod_{i=1}^{89} \tan i^° = \tan 1^° \cdot \tan 2^° \cdots \tan 89^° $$

Answer 1

HINT:

$$\tan(90^\circ-x)=\cot x=\frac1{\tan x}$$

Set $\displaystyle x=1^\circ,2^\circ,\cdots43^\circ,44^\circ$

or $\displaystyle x=46^\circ,47^\circ,\cdots88^\circ,89^\circ$

Answer 2

Another approach is to use the product-to-sum formula $$\tan a\tan b=\frac{\cos(a-b)-\cos(a+b)}{\cos(a-b)+\cos(a+b)}$$

You can pair up factors in the product where the angles sum to 90 degrees --- 1 and 89, 2 and 88, and so on --- and you get $\cos90=0$ in the numerator and denominator, so that each pairing will take the form of $\cos(a-b)$ over $\cos(a-b)$. This pairing leaves you with a bunch of ones together with $\tan45$, which is also 1. So the product is 1.