PreCalc 2 · Analytic Trigonometry
Show that the two sides are equal. Start on one side and use a reciprocal, quotient, Pythagorean, co-function, or even or odd identity.
Fundamental identities
These reciprocal identities are also called the fundamental identities.
csc θ = 1sin θ
sec θ = 1cos θ
cot θ = 1tan θ
Quotient identities
tan θ = sin θcos θ
cot θ = cos θsin θ
Pythagorean identities
sin² θ + cos² θ = 1
1 + tan² θ = sec² θ
1 + cot² θ = csc² θ
Co-function identities
sin(90° − θ) = cos θ
cos(90° − θ) = sin θ
tan(90° − θ) = cot θ
The same statements in radians use π2 − θ.
Even and odd identities
sin(−θ) = −sin θ
cos(−θ) = cos θ
tan(−θ) = −tan θ
Sine and tangent are odd. Cosine is even.
How to verify
Start with one side.
Write the identity you use.
Substitute, then simplify.
Stop when that side matches the other side.
One identity takes you to the other side.
csc θ = 1sin θ
sin θ · 1sin θ
1
Answer: 1
sec θ = 1cos θ
cos θ · 1cos θ
1
Answer: 1
cot θ = 1tan θ
tan θ · 1tan θ
1
Answer: 1
sin θcos θ · cos θ
sin θ
Answer: sin θ
cos θsin θ · sin θ
cos θ
Answer: cos θ
sin² θ + cos² θ = 1
sin² θ = 1 − cos² θ
1 − cos² θ = sin² θ
Answer: sin² θ
1 + tan² θ = sec² θ
sec² θ − 1 = tan² θ
Answer: tan² θ
1 + cot² θ = csc² θ
csc² θ − cot² θ = 1
Answer: 1
sin(−θ) = −sin θ
−sin θ + sin θ
0
Answer: 0
cos(−θ) = cos θ
cos θ − cos θ
0
Answer: 0
sin(90° − θ) = cos θ
cos θ − cos θ
0
Answer: 0
Use two identities, or simplify a fraction.
sin² θ + cos² θ = 1
1 − sin² θ = cos² θ
cos² θ · sec θ
sec θ = 1cos θ
cos² θ · 1cos θ
cos θ
Answer: cos θ
1 − cos² θ = sin² θ
sin² θsin θ
sin θ
Answer: sin θ
sec² θ − 1 = tan² θ
tan² θtan θ
tan θ
Answer: tan θ
cot θ = cos θsin θ
sec θ = 1cos θ
sin θ · cos θsin θ · 1cos θ
1
Answer: 1
1 + tan² θ = sec² θ
sec² θ = 1cos² θ
1cos² θ · cos² θ
1
Answer: 1
tan(−θ) = −tan θ
−tan θ + tan θ
0
Answer: 0
cos(−θ) = cos θ
cos θ · sec θ
sec θ = 1cos θ
cos θ · 1cos θ
1
Answer: 1
sin(90° − θ) = cos θ
cos θ · sec θ
sec θ = 1cos θ
cos θ · 1cos θ
1
Answer: 1
cos(π2 − θ) = sin θ
sin θ · csc θ
csc θ = 1sin θ
sin θ · 1sin θ
1
Answer: 1
csc² θ − 1 = cot² θ
cot² θcot θ
cot θ
Answer: cot θ
cos(−θ) = cos θ
tan θ · cos θ
tan θ = sin θcos θ
sin θcos θ · cos θ
sin θ
Answer: sin θ
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