# Are undefined constants valid in trig equations when other constants present?

1. Jul 22, 2012

### Sduibek

For example:

tan(pi/2 - pi/4) = (tan(pi/2) - tan(pi/4) ) / 1 + tan(pi/2)tan(pi/4)

Which of course comes out to:

undefined + 1 / 1 + undefined

Does that equal 1, or equal Undefined/No Solution?

sorry for the poor formatting, I couldn't find the mathprint symbols for pi and fractions

2. Jul 22, 2012

### LCKurtz

Trig identities are only valid if the various quantities are defined. So, for example, the formula$$\tan(a-b) = \frac {\tan a - \tan b}{1 + \tan a \tan b}$$does not apply if either of the angles or their difference is an odd multiple of $\pi/2$. [Edited, thanks for catching that Curious3141]

Last edited: Jul 22, 2012
3. Jul 22, 2012

### Sduibek

Okay, thank you!

4. Jul 22, 2012

### Curious3141

(Just to correct a small error, the expression *is* perfectly valid if either or both of the angles is an even multiple of $\frac{\pi}{2}$).

Whilst this is strictly true (for odd multiples of $\frac{\pi}{2}$), one can actually use the angle sum identities to calculate the limiting values of expressions.

For example, we know that $\tan (\frac{\pi}{2} - x) = \cot x$

We can prove this easily with a right triangle, but we can also view this as the limit of this expression:

$\lim_{y \rightarrow \frac{\pi}{2}} \tan (y - x) = \lim_{y \rightarrow \frac{\pi}{2}} \frac{\tan y - \tan x}{1 + \tan x\tan y}$ where $x \neq \frac{(2n+1)\pi}{2}$.

Since approaching the limit, $|\tan y| >> |\tan x|$, the limit becomes:

$\lim_{y \rightarrow \frac{\pi}{2}} \frac{\tan y}{\tan x\tan y} = \frac{1}{\tan x} = \cot x$, as expected.

Usually, when I do rough work "informally", I don't bother with the limits, I just put the values as $\tan \frac{\pi}{2}$ directly. But of course, I'm aware this is mathematically incorrect, just a convenient shorthand.

Last edited: Jul 22, 2012