Quite often, I need to verify that two expressions are equivalent (see example below). Typically, these expressions are not excessively complicated, involving just some fractions, exponents and the like. What makes these expressions sometimes difficult to deal with is that
- they can be quite large/messy (many variables, many terms, …),
- "nifty" tricks may be required to go from the first to the second expression.
My problem is: Sometimes I am stuck when verifying that two expressions are equivalent – it takes me hours, if not days. What are strategies to solve such problems more effectively?
To clarify, this is not about knowing math rules. I know the rules (I suppose); after all, this is nothing too advanced, from a technical point of view. The problem is more about figuring out which (inherently simple) steps to take in order to get from expression 1 to expression 2.
Of course, one could say that after having learned the rules, figuring out which steps to take is just a matter of practice and experience. However, I think that there are some potential helpful strategies (see below) and I am asking for further or more effective strategies.
Here's an example:
The expression $\left[ \left( \frac{\tau_1}{\tau_2} \right)^{x} \left( \frac{1-\tau_1}{1-\tau_2} \right)^{1-x} \right]^{\frac{a}{1-a}}$ is supposed to be equivalent to $\left[ 1 + \frac{\Delta^{a}}{\tau_2 \left( 1-\Delta^{a}\right)} \right]^{\frac{a x}{1-a}} \left(1-\Delta^{a}\right)^{\frac{a}{1-a}}$, using $\tau_1 \equiv \Delta^{a} + \tau_2 \left(1-\Delta^{a}\right)$.
Upon plugging in for $\tau_1$, this yields:
$\left[ \left( \frac{\Delta^{a} + \tau_2 \left(1-\Delta^{a}\right)}{\tau_2} \right)^{x} \left( \frac{1-(\Delta^{a} + \tau_2 \left(1-\Delta^{a}\right))}{1-\tau_2} \right)^{1-x} \right]^{\frac{a}{1-a}} \overset{?}{=} \left[ 1 + \frac{\Delta^{a}}{\tau_2 \left( 1-\Delta^{a}\right)} \right]^{\frac{a x}{1-a}} \left(1-\Delta^{a}\right)^{\frac{a}{1-a}}$.
Strategies I commonly employ already:
- Enter the formulas into some software package (like Matlab or – in my case – R) and plug in some arbitrary numbers for the variables (within the admissible range for the respective variables) to see whether both expressions evaluate to the same value. Although this is not helpful in finding a path from the first to the second expression, it helps to identify cases where both are in fact not equivalent. (Leaving aside potential numerical inaccuracies.)
- For large expressions, I try to focus on them "piece by piece", e.g. first simplifying a messy denominator before doing something else with the whole fraction.
- If some messy sub-expression pops up repeatedly, I introduce auxiliary variables defined as equivalent to the sub-expression to "clean up" the overall term by substituting the messy sub-expression. In the example, $\tau_2(1-\Delta^a)$ could be a candidate for this.
- I try to think about the structure of "expression 2" in order to figure out which transformations to apply to "expression 1". In my example this lets me try to factor out $(1-\Delta^a)^{\frac{a}{1-a}}$ from the first expression, because this matches the structure of expression 2.
- Either expand as many expressions as possible or factor out as much as possible, turning $\Delta^{a} + \tau_2 \left(1-\Delta^{a}\right)$ into $\Delta^{a} + \tau_2 - \tau_2 \Delta^{a}$ or $\tau_2 + \Delta^a(1-\tau_2)$ (not helpful in this case).
- Write fractions as negative exponents or vice versa (not helpful in my example).
- Split fractions, like $\frac{1-(\Delta^{a} + \tau_2 \left(1-\Delta^{a}\right))}{1-\tau_2} = \frac{\Delta^{a} + \tau_2 \left(1-\Delta^{a}\right)}{\tau_2-1} - \frac{1}{\tau_2 -1}$. Looks kind of promising because the first fraction is very similar to the $(\dots)^x$ term, except for the denominator. However, this still leaves me with a sum raised to the power of $1-x$ …
- [Random other stuff you can do with fractions and exponents.]
- Take a break / walk or employ rubber-duck-debugging-like strategies (like posting on math.SE).
- Try to be creative …
- Work using different media like a sheet of paper, a whiteboard or typing.
Although I'm definitely interested in a solution to my sample problem, I'm mostly looking for advice on how to tackle such problems in general.