



An identity is an equation which is obviously true.







Like: 


X
= X 
3
= 3 
1
hour = 60 minutes 
6
= ^{1}/2 dozen 






We have
already seen a couple of trig identities in other chapters: 





sin^{2}X 
+ 
cos^{2}X 
= 
1 





tan
X 
= 
sin
X 



cos
X 





sec
X 
= 
1 



cos
X 





csc
X 
= 
1 



sin
X 





cot
X 
= 
1 
= 
cos
X 


tan
X 
sin
X 






In trig
identity problems, you get some messy trig function equation. 


You are asked
to verify that the left side of the equation 


DOES equal the right
side of the equation. 





You might be
given an equation like: 





cosXtanXcscX
= 1 





The strategy
is to change the more complicated side of the equation 


to make
it look like the
simpler side. 


There is no
exact strategy for solving these . 


Each problem
is a puzzle with it's own route to the solution. 





What is
generally called the "easy" way to solve these things 


is
to substitute for tan, sec, csc, and cot 


so that the only terms are
sine and cosine. 





That means
using the identities we listed above. 





The
substitutions would go like this: 





cosXtanXcscX
= 1 





cosX 
tanX 
cscX 
= 
1 





cosX 
sinX 
1 
= 
1 



1 
cosX 
sinX 






cosX 
sinX 
= 
1 




cosX 
sinX 







1 
= 
1 










This strategy
works fine, and if you like it, great. 





Let's try
another one. 





1 
+ 
tan^{2}X 
= 
csc^{2}X 

tan^{2}X 






Break it up
into sines and cosines... 





1 
+ 
sin^{2}X 
= 
1 

cos^{2}X 



sin^{2}X 

sin^{2}X 




cos^{2}X 







Generally,
when we have something like 1 ± a fraction, 


you want to turn it
into one term. 





To do this you
need to have a common denominator and another way to say 1. 


We can write
the 





1 
+ 
sin^{2}X 
part
of the equation as 
cos^{2}X 
+ 
sin^{2}X 



cos^{2}X 
cos^{2}X 
cos^{2}X 






Now we have a
common denominator. 


Put this piece
back in the equation. 





cos^{2}X 
+ 
sin^{2}X 
= 




cos^{2}X 
cos^{2}X 
1 



sin^{2}X 

sin^{2}X 





cos^{2}X 









cos^{2}X
+ sin^{2}X 
= 





cos^{2}X 
1 




sin^{2}X 
sin^{2}X 




cos^{2}X 







Now we get to
the "Nitty Gritty." 


There are two
ways you can legally "blow away" trig terms in these
equations 


One is
canceling (like we did in the first example). 


The other is
using an identity like: 





<some trig stuff> = 1 





Then we just put in the 1 for the trig stuff. 


The one
totally important trig identity there is goes like this: 





sin^{2}X
+ cos^{2}X = 1 





Look at what
we have on the top left! 


We make that
substitution. Take out the trig stuff. Put in the 1 and we get ... 






1 
= 





cos^{2}X 
1 




sin^{2}X 
sin^{2}X 




cos^{2}X 







Now we
multiply the left side of the equation 


by another name for 1 to make
things less messy. 


That name for
1 is cos^{2}X/1 divided by itself. 


We get: 


cos^{2}X 
X 
1 
= 




1 
cos^{2}X 
1 



cos^{2}X 
sin^{2}X 
sin^{2}X 



1 
cos^{2}X 







Multiplying and simplifying: 




cos^{2}X
^{1} 
= 







cos^{2}X
^{1} 
1 






cos^{2}X
^{1} sin^{2}X 
sin^{2}X 






cos^{2}X
^{1} 













This all
works, and like I said, 


if you are comfortable doing algebra on trig
functions 


like we've just done you will have no problem with these
things. 





Many people
who did really well in algebra 


have trouble getting used to doing
algebra on these trig function things. 





For these
people (and I was one of them), 


there is another way to work these
problems. 





When we
started graphing triangles, 


we built a right triangle with a
hypotenuse length of 1. 











The sine of
angle "A" is opposite over hypotenuse, that is: 











The cosine of
angle "A" is adjacent over hypotenuse, that is: 











The tangent of
angle "A" is opposite over adjacent, that is: 











So we have: 





sin
A (or sin X or sin
(whatever)) = Y 


cos
A (or cos X or cos
(whatever)) = X 


tan
A (or tan X or tan
(whatever)) = 
Y 

X 






The other
three trig functions are just ^{1}/_{(these three) }so: 























Since cos
^{2}X
+ sin ^{2}X = 1, we can also say that X ^{2} + Y ^{2}
= 1 





Now, we can substitute in some form of X's and Y's for all trig functions. 


We might find that makes it easier to solve these things. 





Let's redo the two examples that way. 





cosX 
tanX 
cscX 
= 
1 





X 
Y 
1 
= 
1 


X 
Y 







XY 
= 
1 





XY 







1 
= 
1 






OK, that wasn't too bad. 


Now let's do the toughie: 





1 
+ 
tan^{2}X 
= 
csc^{2}X 

tan^{2}X 






1 
+ 
Y^{2} 
= 
1 

X^{2} 


Y^{2} 
Y^{2} 

X^{2} 






Turn the 1 at
the top left into X ^{2}/X ^{2} and combine terms: 





X^{2 + }Y^{2} 
= 
1 

X^{2} 


Y^{2} 
Y^{2} 

X^{2} 






Use the X
^{2} + Y ^{2} = 1 thing to simplify: 





1 
= 
1 

X^{2} 


Y^{2} 
Y^{2} 

X^{2} 






Multiply the left side by a fancy name for 1 





X^{2} 
x 
1 
= 
1 


1 
X^{2} 



X^{2} 
Y^{2} 
Y^{2} 


1 
X^{2} 






Combine terms
and simplify: 







X^{2}^{
1} 
= 
1 



X^{2}^{
1} 




X^{2}Y^{2} 
Y^{2} 



X^{2}^{
1} 












We can now
take this back to where the X's and Y's came from as: 





csc^{2}X
= csc^{2}X 





The steps we
took with the X's and Y's 


are actually the same steps we took with
the trig functions. 





copyright 2008 Bruce Kirkpatrick

