Download Proving Identities

Proving Identities
A trigonometric identity is a statement that is true for all values of the variable
for which it is defined. In order to prove a trigonometric identity, we use
algebraic manipulations and trigonometric substitutions, based on identities that
we already know, to transform the left side of the identity into the right-hand
side or to transform the right-hand side of the identity into the left-hand side.
We work on each side separately. We do not use properties from algebra that
involve both sides of the identity, such as multiplying both sides by the same
quantity.
You have a list in the class notes on Basic Identities.
EXAMPLE 1: Prove that tan x(cos x  cot x)  sin x  1 .
Solution: We will simplify the left-hand side and leave the right-hand side as it is.
First, we will distribute the factor tan x on the left-hand side:
tan x cos x  tan x cot x  sin x  1
sin x
cos x
Next, we will rewrite tan x 
and cot x 
(Ratio Identities):
cos x
sin x
sin x
sin x cos x
 cos x 

 sin x  1
cos x
cos x sin x
Finally, we will cancel common factors of the numerator and denominator,
leaving:
sin x  1  sin x  1
The identity is proved.
cos 4 t  sin 4 t
EXAMPLE 2: Prove that
 csc 2 t  sec 2 t .
2
2
sin t cos t
Solution: Again, we will start by simplifying the left-hand side.
First, we can factor the numerator as a difference of squares:
(cos 2 t  sin 2 t )(cos 2 t  sin 2 t )
 csc 2 t  sec 2 t
2
2
sin t cos t
Next, we use a Pythagorean Identity: cos 2 t  sin 2 t  1:
1  (cos 2 t  sin 2 t )
 csc 2 t  sec 2 t
2
2
sin t cos t
Now we will divide each term in the numerator by the product sin 2 t cos 2 t :
cos 2 t
sin 2 t

 csc 2 t  sec 2 t
2
2
2
2
sin t cos t sin t cos t
1
1

 csc 2 t  sec 2 t
2
2
sin t cos t
Finally, we use Reciprocal Identities to show that the left-hand side is the same
as the right-hand side;
csc 2 t  sec 2 t  csc 2 t  sec 2 t
The identity is proved.
EXAMPLE 3: Prove that
1  cos 
sin 
.

sin 
1  cos 
Solution: This time, since the denominator on the right-hand side is more
complicated, we will start by simplifying the right-hand side. In order to
transform the expression on the right to the one on the left, we need to convert
the denominator to an expression with only one term. We will use a technique
similar to one you may have used in Algebra to rationalize denominators of
fractions.
We will multiply the numerator and denominator on the right by 1 cos 
(which is like the conjugate of the denominator):
1  cos 
sin  1  cos 


sin 
1  cos  1  cos 
Now we will expand the product in the denominator:
1  cos  sin  (1  cos  )

sin 
1  cos 2 
Next, we replace 1 cos 2  by sin 2  (Pythagorean Identity):
1  cos  sin  (1  cos  )

sin 
sin 2 
Finally, we cancel one factor of sin  from the numerator and denominator,
which proves the identity:
1  cos  1  cos 
.

sin 
sin 
EXAMPLE 4: Prove that
sin 2 x  tan 2 x
 sin 2 x .
1  sec 2 x
Solution: The left-hand side is more complicated, so we will work on that side.
We first use a Pythagorean Identity to replace 1  sec 2 x  (sec 2 x  1) by
 tan 2 x :
sin 2 x  tan 2 x
 sin 2 x
2
 tan x
Next, we can divide each term in the numerator by  tan 2 x :
sin 2 x tan 2 x


 sin 2 x
2
2
tan x tan x
In the next steps, we simplify the fractions (In the first fraction, we use a
sin 2 x
Reciprocal Identity to replace tan 2 x by
.):
cos 2 x
sin 2 x

 1  sin 2 x
2
 sin x 


2
 cos x 
cos 2 x
 1  sin 2 x
2
sin x
2
 cos x  1  sin 2 x
Finally, we use a Pythagorean Identity to replace  cos 2 x  1  1  cos 2 x by
sin 2 x , and the identity is proved:
sin 2 x  sin 2 x .
 sin 2 x 
EXAMPLE 5: Graph the left expression and the right expression together (using
your graphing calculator). If the graphs appear to be the same, then the
equation is probably an identity, and you should try to prove the identity. If the
graphs do not appear to be the same, prove that the statement is not an identity
by finding a specific counterexample.
sin x  1  cos x
Solution: The graph below shows y  sin x in blue and y  1 cos x in black.
2
1.5
1
0.5
2
2
-0.5
-1
It is clear that the graphs are different and that this equation is not an identity.
Specifically, if x   , for example, the left-hand side of the equation equals
sin x  sin   0 , but the right-hand side of the equation equals
1  cos x  1  cos   1  (1)  2 , and the two sides of the equation are not the
same.
(This specific value of x for which the two sides of the equation differ is called a
counterexample. You can prove that a statement is false by finding just one
counterexample. However, even one million examples would not prove that an
identity is true! To prove that an identity is true, you need a logical argument
like the ones provided in the examples.)