A trigonometric identity is an equation involving trigonometric
functions that holds true for all angles. These identities are commonly
called Pythagorean identities, because they come from the Pythagorean
In general, the Math IIC will test your knowledge of the
trigonometric identities by giving you a complex expression that
you have to simplify. These questions have more to do with memorizing
the identities and being good with algebraic substitution than they
do with the theoretical concepts of trigonometry.
To solve a problem like this, use the trigonometric identities
to simplify the trigonometric expression into sines and cosines.
After you have simplified the expression using the identities, it
is quite likely that the expressions will simplify further through
the canceling of terms. The simplification of the expression in
the example questions proceeds as follows.
Simplify the mess given to you by the problem, and you
Here’s another example:
is tan2 cos2 + 1 – sin2?
Other Trigonometric Identities
About once a test, you may encounter a particularly complicated
trigonometric expression. Such expressions can usually
be solved using the trigonometric identities we’ve already covered,
but the process will be laborious and difficult. However, if you
know the special trigonometric identities below, these questions
will be much easier and less time-consuming.
We list the special trigonometric identities in the order
of usefulness, starting with the ones most likely to be used on
Sum and Difference Formulas
There are three identities for the sum of two angles:
There are also three identities for the difference of
Try to use these identities in a sample problem:
the following expression: sin2x +
cos2x – sec2x +
Using the identities sin2x +
cos2x = 1 and 1 + tan2x =
sec2x, the expression
simplifies in the following way:
Here’s another example to practice with:
sin a = –cos b =
3/5 and a and b are
both in the second quadrant, what is cos (a – b)?
If sin a =
the angle is in the second quadrant, then cos a =
Also, if cos b = –3
/5 and b is
in the second quadrant, then sin b =
See the figure below:
Now you can use the subtraction identity
to see that
If you are familiar with the formula for cos(a
and remember the Pythagorean triple 3-4-5, you can save time on
a question like this. Otherwise, you have to calculate the arcsine
the arccosine of –3/
in order to find
, respectively. Then you have to
and take the cosine of the resulting
Double Angle Formulas
There are three double angle formulas.
Half Angle Formulas
Of all trigonometric identities, the six half-angle formulas
appear most infrequently on the Math IIC.