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What is simpler, $\tan^2\theta+1$ or $\sec^2\theta$? I prefer to put trig expressions exclusively in terms of $\sin$, $\cos$, and $\tan$, but the second expression is shorter.

What is simpler, $$x^6-20x^5+148x^4-518x^3+907x^2-758x+240$$ or

$$(x-1)(x-1)(x-2)(x-3)(x-5)(x-8)?$$

You may want your polynomials expressed in terms of the standard basis $1,x,x^2,\ldots$, or factored into linear terms; or then again, expressed in Bernstein form.

The solution in each case depends on what is required from the expression, so the answer to your question is that "simplify" is not well defined. What happens in practice is that instructors in remedial math courses teach some simplification rules so the students obtain a canonical answer that can be compared to the answer at the back of the textbook... And what dumb rules they are sometimes! I particularly mind that students learn to write $\sin(\pi/4) = \frac{\sqrt{2}}{2}$ instead of the "simpler" $\frac{1}{\sqrt{2}}$ (apparently some authors think that the students will be scared if there is a radical in the denominator). As a result the students learn not to think by themselves.

But all of this is moot... the true answer is that "simplifying" an expression may not be practical if, for instance, your expression is a word representing an element of a group whose word problem is not solvable :)

What is simpler, $1+\tan^2\theta$ or $\sec^2\theta$? I prefer to put trig expressions exclusively in terms of $\sin$, $\cos$, and $\tan$, but the second expression is shorter.

What is simpler, $$x^6-20x^5+148x^4-518x^3+907x^2-758x+240$$ or

$$(x-1)(x-1)(x-2)(x-3)(x-5)(x-8)?$$

You may want your polynomials expressed in terms of the standard basis $1,x,x^2,\ldots$, or factored into linear terms; or then again, expressed in Bernstein form.

The solution in each case depends on what is required from the expression, so the answer to your question is that "simplify" is not well defined. What happens in practice is that instructors in remedial math courses teach some simplification rules so the students obtain a canonical answer that can be compared to the answer at the back of the textbook... And what dumb rules they are sometimes! I particularly mind that students learn to write $\sin(\pi/4) = \frac{\sqrt{2}}{2}$ instead of the "simpler" $\frac{1}{\sqrt{2}}$ (apparently some authors think that the students will be scared if there is a radical in the denominator). As a result the students learn not to think by themselves.

But all of this is moot... the true answer is that "simplifying" an expression may not be practical if, for instance, your expression is a word representing an element of a group whose word problem is not solvable :)

What is simpler, $1+\tan^2\theta$ or $\sec^2\theta$? I prefer to put trig expressions exclusively in terms of $\sin$, $\cos$, and $\tan$, but the second expression is shorter.

What is simpler, $$x^6-20x^5+148x^4-518x^3+907x^2-758x+240$$ or

$$(x-1)(x-1)(x-2)(x-3)(x-5)(x-8)?$$

You may want your polynomials expressed in terms of the standard basis $1,x,x^2,\ldots$, or factored into linear terms; or then again, expressed in Bernstein form.

The solution in each case depends on what is required from the expression, so the answer to your question is that "simplify" is not well defined. What happens in practice is that instructors in remedial math courses teach some simplification rules so the students obtain a canonical answer that can be compared to the answer at the back of the textbook... And what dumb rules they are sometimes! I particularly mind that students learn to write $\sin(\pi/4) = \frac{\sqrt{2}}{2}$ instead of the "simpler" $\frac{1}{\sqrt{2}}$ (apparently some authors think that the students will be scared if there is a radical in the denominator). As a result the students learn not to think by themselves.

But all of this is moot... the true answer is that "simplifying" an expression may not be practical if, for instance, your expression is a word representing an element of a group whose word problem is not solvable :)

What is simpler, $\tan^2\theta+1$ or $\sec^2\theta$? I prefer to put trig expressions exclusively in terms of $\sin$, $\cos$, and $\tan$, but the second expression is shorter.

What is simpler, $$x^6-20x^5+148x^4-518x^3+907x^2-758x+240$$ or

$$(x-1)(x-1)(x-2)(x-3)(x-5)(x-8)?$$

You may want your polynomials expressed in terms of the standard basis $1,x,x^2,\ldots$, or factored into linear terms; or then again, expressed in Bernstein form.

The solution in each case depends on what is required from the expression, so the answer to your question is that "simplify" is not well defined. What happens in practice is that instructors in remedial math courses teach some simplification rules so the students obtain a canonical answer that can be compared to the answer at the back of the textbook... And what dumb rules they are sometimes! I particularly mind that students learn to write $\sin(\pi/4) = \frac{\sqrt{2}}{2}$ instead of the "simpler" $\frac{1}{\sqrt{2}}$ (apparently some authors think that the students will be scared if there is a radical in the denominator). As a result the students learn not to think by themselves.

But all of this is moot... the true answer is that "simplifying" an expression may not be practical if, for instance, your expression is a word representing an element of a group whose word problem is not solvable :)