## Monday, April 9, 2007

### Get A Tan. Topic: Trigonometry. Level: AMC.

Problem: (2007 MAO State - Gemini) Let $x$ be in degrees and $0^{\circ} < x < 45^{\circ}$. Solve for $x$: $\tan{(4x)} = \frac{\cos{x}-\sin{x}}{\cos{x}+\sin{x}}$.

Solution: Here's a nice tangent identity that is not very well-known, but rather cool. Start with the regular tangent angle addition identity,

$\tan{(x+y)} = \frac{\tan{x}+\tan{y}}{1-\tan{x} \cdot \tan{y}}$.

Letting $x = 45^{\circ}$ and $y = -\theta$, we obtain

$\tan{(45^{\circ}-\theta)} = \frac{1-\tan{\theta}}{1+\tan{\theta}} = \frac{\cos{\theta}-\sin{\theta}}{\cos{\theta}+\sin{\theta}}$.

Well, look at that. It's the same expression as the RHS of the equation we want to solve. Substituting accordingly, it remains to solve

$\tan{(4x)} = \tan{(45^{\circ}-x)} \Rightarrow 4x = 45^{\circ}-x \Rightarrow x = 9^{\circ}$.

QED.

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Comment: This identity is a nice one to keep around because it can turn up unexpectedly. Especially when you see that exact form and you're like "whoa this is such a nice form there must be an identity for it." So there.

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Practice Problem: (2007 MAO State - Gemini) One hundred positive integers, not necessarily distinct, have a sum of $331$. What is the largest possible product these numbers can attain?