Download 5.1

Verifying Trig
Identities
(5.1)
JMerrill, 2009
(contributions from DDillon)
Trig Identities
Identity: an equation that is true for
all values of the variable for which
the expressions are defined
Ex:
sin x
tan x 
cos x
or (x + 2) = x + 2
Conditional Equation: only true for
some of the values
Ex: tan x = 0
or x2 + 3x + 2 = 0
Recall
y
sin  
r
x
cos  
r
y
tan  
x
r
csc  
y
r
sec  
x
x
cot  
y
Recall - Identities
Reciprocal Identities
sin  
1
csc
1
cos 
sec
1
tan  
cot 
Also true:
1
csc 
sin 
1
sec 
cos
1
cot  
tan 
Recall - Identities
Quotient Identities
sin 
tan  
cos
cos
cot  
sin 
Fundamental
Trigonometric Identities
Negative Identities (even/odd)
sin      sin 
csc      csc
cos     cos
sec     sec
tan      tan 
cot      cot 
These are the only
even functions!
Recall - Identities
Cofunction Identities


sin   cos    
2



cos  sin    
2



tan   cot    
2



cot   tan    
2



sec  csc    
2



csc  sec    
2

Recall - Identities
Pythagorean Identities
sin   cos   1
2
2
1  cot   csc 
2
2
tan   1  sec 
2
2
Simplifying Trig
Expressions
• Strategies
• Change all functions to sine and cosine (or at
least into the same function)
• Substitute using Pythagorean Identities
• Combine terms into a single fraction with a
common denominator
• Split up one term into 2 fractions
• Multiply by a trig expression equal to 1
• Factor out a common factor
Simplifying # 1
cot x sin x
cos x

sin x
sin x
cos x

sin x
sin x
 cos x
2
Simplifying #2
cos x
 sin x
sin x
cos 2 x sin 2 x


sin x
sin x
2
2
cos x  sin x

sin x
1

sin x
 csc x
Simplifying #3
1  cos x
2
cos x
2
2
sin x

2
cos x
 tan x
2
Simplifying #4
cos x  sin x tan x
sin x
 cos x  sin x
cos x
2
sin x
 cos x 
cos x
2
2
cos x sin x


cos x cos x
1

cos x
 sec x
Simplifying #5

 2
sin x  cos   x  cos x
2

3
 sin x  sin x cos x
3
2
 sin x  sin x  cos x 
2
 sin x
2
Proof Strategies
• Never cross over the equal sign (you
cannot assume equality)
• Transform the more complicated side of
the identity into the simpler side.
• Substitute using Pythagorean identities.
• Look for opportunities to factor
• Combine terms into a single fraction with a
common denominator, or split up a single
term into 2 different fractions
• Multiply by a trig expression equal to 1.
• Change all functions to sines and cosines,
if the above ideas don’t work.
ALWAYS TRY SOMETHING!!!
Example
sin 
cos 

 csc 
1  cos  sin 
• Prove
• 2 fractions that need to be added:
• Shortcut:  sin   sin     cos   1  cos  
1  cos    sin  
sin2   cos   cos2 
1  cos    sin  
1  cos 
1  cos   sin 
1

 csc 
sin 
Show cos x 1  cot x   cot x
2
2
2
cos 2 x 1  cot 2 x 
1 + cot2x = csc2 x
cos x  csc x 
1
csc x 
2
sin x
2
2
2
1
cos x 2
sin x
2
cos 2 x
sin 2 x
 cot x
2
tan x  cot x
Prove
 tan x
2
csc x
tan x  cot x
csc 2 x
sin x cos x

cos x sin x
csc 2 x
sin x cos x

cos x sin x
2
csc x
sin 2 x  cos 2 x
sin x cos x
csc2 x
1
sin x cos x
1
sin 2 x
 sin 2 x 
1


sin x cos x  1 
sin x
cos x
 tan x