How should trigonometric expressions be simplified?

Question

I have been learning trigonometric identities and I am having trouble understanding how they should be applied to simplify expressions. For example, the expression $2\sin{x} \cos{x}$ is equal to $\sin{2x}$. But which of the two expressions is “simpler”? While the second is decidedly shorter, it also contains a multiplication nested in a sine function. Is there a general rule for characteristics that should be avoided in an expression (e.g. exponents, nested formulas, radicals$\ldots$)?

Answer 1

A general rule is that you want the fewest number of special functions ($\sin, \cos, \exp, \dots$). It's not less "simple" to nest basic math like multiplication and radicals, but nesting special functions should be avoided. Off course there are no real rules, so you'll have to decide for yourself what's easier to read.

Answer 2

That depends.

For instance, one would usually argue that $\cos^5 x$ is simpler than

$$\frac1{16}(\cos 5x+5\cos 3x+10\cos x)$$

However, in the context of integration, the latter is actually simpler (they are always equal, in case you wonder).

It's really a matter of context, and of experience you'll acquire.

Then there are also habits. I have had teachers that viewed $\frac{\sqrt{2}}2$ as simpler than $\frac1{\sqrt2}$, just because the square root is on the numerator. I have never really agreed with that. It is simpler only if you compute its numerical value by hand. Something that is not likely to happen these days.