Listen to what we say
In general, the strategies to prove
an identity are as follows.
- Find the differences in angles,
operations and functions of the two sides of the equation.
- Decide which difference you want
to reduce first. You might find you are reducing more than one
difference at a time while you work. That's fine.
- Think about what kind of formulas
that can be used to reduce the differences, or what kind of
strategies you can use. Choose appropriate formulas to reduce the
differences. Remember that you can use formulas from either side.
As long as you are reducing the differences, you are on the right
way. If you find you are not reducing the differences, you need be
careful that you know what you are going to get.
- Comparing the result you get at
every step with the other side of the identity. Find out what the
differences are, and continue the verification.
- For conditional identities, you
should use all the conditions, and compare the two sides of the
goal identity, which you need to prove. You should also compare the
goal identity with the known conditional identity if there are any.
If you can't go farther when you are proving the identity, it could
be the time for you to use the conditions.
- When all the differences of the
two sides of the equation are canceled, then the verification is
While you work more and more on
problems, using our idea, you will find more and more strategies.
You might even solve some problems you can't ever imagine, either
in mathematics or in the real world. We honestly wish you to have
fun while doing them.
Now, you have finished our Examples
section, do you understand our ideas? Do you know all the formulas?
Do you understand our examples and know why we took each step? We
hope that you very well understand how to discover some clues in a
problem and how to choose some formulas to solve the problem by
using the clues. If so, we are very happy to congratulate you that
you have finished our Learning sections.