Transcript 7.1 Trig Identities

7.1 Trig Identities
• Simplifying Trig Expressions
• Proving Trig Identities
Fundamental Trig Identities
Reciprocal Identities:
Tangent/Cotangent Identities:
1
sin x
1
sec x 
cos x
csc x 
cot x 
sin x
cos x
cos x
cot x 
sin x
tan x 
1
tan x
Pythagorean Identities:
sin 2 x  cos2 x  1
tan 2 x  1  sec2 x
1  cot 2 x  csc2 x
Example 1
Simplify the trig expression: cos t  tan t sin t
Solution:
 sin t 
cos t  tan t sin t  cos t  
 sin t
 cos t 
sin 2 t
 cos t 
cos t
cos 2 t  sin 2 t

cos t
1

cos t
 sect
Example 2
Simplify the expression: sin u  cot u cos u
Answer:
cscu
Example 3
Simplify the expression:
Solution:
cos x
cos x

1  sin x 1  sin x
cos x
cos x
cos x(1  sin x)
cos x(1  sin x)



1  sin x 1  sin x
(1  sin x)(1  sin x) (1  sin x)(1  sin x)

cos x  sin x cos x cos x  sin x cos x

2
1  sin x
1  sin 2 x
2cos x

1  sin 2 x
2cos x

cos2 x

2
cos x

2sec x
Tips for Proving Trig Identities




Start with one side of the equation and manipulate it until it
equals the other side. (Try the more complicated side first!)
Look for chances to use identities and/or algebraic
techniques (adding fractions, factoring, multiplying by a form
of “1”, etc.)
If you get stuck, try re-writing everything in terms of the
sine and cosine.
* Can also try working with each side of the equation
separately until you obtain the same expression.
Example 4
1
1
Prove the identity: 2 tan x sec x 

1  sin x 1  sin x
Example 5
Verify the identity:
1  cos 
tan 2 

cos 
sec   1
1  cos 
1
cos 
 sec   1


LHS =
cos  cos 
cos 
tan 2 
sec2   1 (sec   1)(sec   1)
 sec   1

RHS =

sec   1
sec   1
sec  1