<s>[INST] Calculating $\tan15\cdot \tan 30 \cdot \tan 45 \cdot \tan 60 \cdot \tan 75$  What is $\tan15\cdot \tan 30 \cdot \tan 45 \cdot \tan 60 \cdot \tan 75$  equal to (in degrees)?
Here is how I tried to solve it:
I assumed $x = \tan 15 (x= 0.27)$, so I rewrote it as:
$x\cdot 2x \cdot 3x \cdot 4x \cdot 5x = 120x^5$
$0.27^5 = 0.001323$
$120 \cdot 0.001323 = 0.16$
But Google Calculator gives me  $-1.19576279$
What is the right way to calculate this  trig expression?
 [/INST] Note that if $0\leq \alpha\leq \frac{\pi}{2}$ ($0^{\circ}$ to $90^{\circ}$), then
$$\cos\left(\frac{\pi}{2}-\alpha\right) = \cos\frac{\pi}{2}\cos\alpha + \sin\frac{\pi}{2}\sin\alpha = \sin\alpha.$$
In terms of degrees, $\cos(90-\alpha) = \sin(\alpha)$. 
So if you look at $\tan(15^{\circ})$ and $\tan(75^{\circ}) = \tan(90^{\circ}-15^{\circ})$, we have:
$$\tan(15)\tan(75) = \frac{\sin(15)}{\cos(15)}\;\frac{\sin(75)}{\cos(75)} = \frac{\sin(15)}{\sin(75)}\;\frac{\sin(75)}{\sin(15)} = 1.$$
Similarly with $\tan 30 \tan 60$. So the entire product is just equal to $\tan(45)$. And $\tan(45^{\circ})$ is equal to...
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