Section 7.1 ­ Trig Identities
Fundamental Identities
Reciprocal Identities
1 csc x = sin x
1 1 sec x = cos x
cot x = tan x
sin x
tan x = cos x
cot x = cos x
sin x
Pythagorean Identities
sin2x + cos2x = 1
1 + tan2x = sec2x
1 + cot2x = csc2x
Even­Odd Identities
sin(­x) = ­ sin x
cos(­x) = cos x
tan(­x) = ­ tan x
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Cofunction Identities
π
sin( ­ x) = cos x
2
π
cos( ­ x) = sin x
2
π
tan( ­ x) = cot x
2
π
cot( ­ x) = tan x
2
π
sec( ­ x) = csc x
2
π
csc( ­ x) = sec x
2
To simplify trig expressions, we use factoring, and common denominators along with the fundamental identities.
Examples: Simplify
sec x ­ cos x
tan x
cos x(sin x)(sec x)
___sin x csc x + cot x
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Guidelines for Proving Trig Identities
1. Start with one side. Your goal is to transform it into the other side. We usually start with the more complicated side.
2. Use known identities. Use algebra and the identities you know to change the side you started with. Find common denominators for fractional expressions, factor and use the fundamental identities to simplify expressions.
3. Convert to sines and cosines. If you are stuck, try converting all functions to sines and cosines.
WARNING!!! TO PROVE AN IDENTITY, WE DO NOT JUST PERFORM THE SAME OPERATIONS ON BOTH SIDES OF THE EQUATION.
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Verify the identity.
cos x + sin x tan x = sec x
tan x cos x = sin x
(tan x + cot x)2 = sec2x + csc2x
1 + cos x ­ 1 ­ cos x = 4cot x csc x
1­ cos x
1 + cos x
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