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Trig Identities
Chapter 3 Section 4
Conditional Equations
• Two functions are said to be identically
equation if f(x) = g(x) for every value of x for
which both functions are defined
• An equation that is not an identity is called a
conditional equation
( x  1) 2  ( x  1)( x  1)  x 2  2 x  1
Trigonometric Formulas
• Quotient Identities
sin  
tan   
cos 
cos 
cot  
sin  
• Reciprocal Identities
1
csc  
sin  
1
sec  
cos 
cot  
1
tan  
Trigonometric Formulas
2


• Pythagorean Identities sin   cos    1
tan 2    1  sec2  
2
cot2    1  csc2  
• Even-Odd Identities
sin      sin  
cos    cos 
tan      tan  
csc     csc 
sec    sec 
cot     cot 
Using Algebra to Simplify Trig Expressions
• Multiplying by “1”
• Write trig expressions over LCD
• Write trig expressions in terms of sine and
cosine only and factoring
– NOTE: Don’t make the final expression more
complicated if it does not make sense to change
to sine and cosine
• When “showing” two identities are equal,
always start with the more complicated side
Simplify
cot  
csc  
Show
csc   tan    sec  
Show
cos  
1  sin  

1  sin  
cos  
Simplify
1  sin  
sin  

cot    cos  
cos  
Simplify
sin 2    1
tan   sin    tan  
Show
sin 2     cos 2   
 cos   sin  
sin     cos  
Show
cot 2 −𝜃 + 1 = csc 2 𝜃
Application
• A searchlight casts a spot of light on a wall located
65 meters from the searchlight.
• The acceleration r is given as
1300 cot 𝜃 [2 cot 2 𝜃 + 1]
• Show this is equivalent to
1300(1+sec2 (𝜃))
tan3 (𝜃)
Show
1300(1 + sec 2 (𝜃))
1300 cot 𝜃 [2 cot 𝜃 + 1] =
tan3 (𝜃)
2
Guidelines
• Always start with the side containing the more
complicated expression
• Rewrite sums or differences as a single
quotient
• If possible, rewrite one side in terms of sine
and cosine only
• Keep Goal in Mind – watch form on other side
of expressions