Trigonometric Identities Reference Sheet
Reciprocal Identities
• sin θ =
1
csc θ
• csc θ =
1
sin θ
• cos θ =
1
sec θ
• sec θ =
1
cos θ
• tan θ =
1
cot θ
• cot θ =
1
tan θ
Quotient Identities
• tan θ =
sin θ
cos θ
• cot θ =
cos θ
sin θ
Pythagorean Identities
• sin2 θ + cos2 θ = 1
• tan2 θ + 1 = sec2 θ (This is just sin2 θ + cos2 θ = 1 divided through by cos2 θ)
• 1 + cot2 θ = csc2 θ (This is just sin2 θ + cos2 θ = 1 divided through by sin2 θ)
Sum & Difference Formulas
• sin (α ± β) = sin α cos β ± cos α sin β (This is two formulas)
• cos (α ± β) = cos α cos β ∓ sin α sin β (This is two formulas; note the change in signs!)
1
Double Angle Formulas
• sin (2θ) = 2 sin θ cos θ (This is just sin (α + β) where you replace both α and β with θ)
• cos (2θ) = cos2 θ − sin2 θ = 1 − 2 sin2 θ = 2 cos2 θ − 1
Even & Odd Identities
sin θ, csc θ, tan θ, & cot θ are all odd functions. In other words:
• sin (−θ) = − sin θ
• csc (−θ) = −cscθ
• tan (−θ) = − tan θ
• cot (−θ) = − cot θ
cos θ & sec θ are even functions. In other words:
• cos (−θ) = cos θ
• sec (−θ) = sec θ
2
Advanced Algebra w/Trig
Trig Identities Worksheet #1
Simplify.
1 − sin 2 θ
sin 2 θ
1.
sin θ (1 + cot 2 θ )
2.
3.
cos θ csc θ
tan θ
4. csc2 α − cot 2 α
5. tan θ csc θ
6.
1
cos 2 θ
−
sin 2 θ sin 2 θ
Verify each identity.
7.
sin x cos x
1
+
=
csc x sec x
Challenge Problem
1 − tan 2 x
9.
= tan 2 x
2
cot x − 1
8.
sin x csc x
= tan x
cot x
Advanced Algebra w/Trig
Trig Identities Worksheet 2
Verify each identity.
1.
tan x
= sin x
sec x
2.
cot x sec x
=1
csc x
3. tan θ sin θ + cos θ =
sec θ
4. cot 2 θ − csc 2 θ =
−1
5. sin x + cos x cot x =
csc x
6.
sec x − cos x
= sin 2 x
sec x
OVER 
Challenge
7.
1 − cot 2 x
= cot 2 x
tan 2 x − 1
Review Material:
Convert the following angles from degrees to radians (leave in terms of pi) or from radians to degrees.
8.
13π
17
9. 72
Find the EXACT value of the following.
10. cos − 405
11. sin180
Advanced Algebra w/Trig
Trig Identities Worksheet 3
Verify each identity.
1.
1 − cos 2 x
= tan 2 x
2
cos x
2.
3.
csc x
= cos x
cot x + tan x
4. sin 2 x − cos 2 x =
1 − 2 cos 2 x
5.
1 − csc x sin x − 1
=
1 + csc x sin x + 1
6.
1 − (sin x − cos x) 2
= 2 cos x
sin x
1 − cos x sec x − 1
=
1 + cos x sec x + 1
OVER 
(sin x − cos x) 2
7. sec x − 2sin x =
cos x
8.
1
cosθ
tan θ
−
=
sin θ cosθ sin θ
Challenge
cos x
cos x
9.
+
=
2sec x
1 − sin x 1 + sin x
Review:
Find the exact value of all θ in [ 0, 2π ) that satisfies the following.
10. sec θ = − 2
11. tan θ = 3
Find the EXACT indicated value given the following information.
12. Find sin θ if sec θ =
26
, and tan θ > 0
24
7
13. Find sec θ if cot θ = − , and sin θ < 0
8
Advanced Algebra w/Trig
Trig Identities REVIEW
Simplify each Expression.
1. cos x  sec x  cos x 
3.
1
1

2
sec  csc2 
Verify each Identity.
sin 2 x  1
2
2
5. tan x  sec x 
cos 2 x
7.
1  cot x sin x  cos x

1  cot x sin x  cos x
2.
1
 sec2 x
2
cot x
4. 1 
sin 
csc 
6.
1
 1  cot 2 x
2
1  cos x
8.
sec x
1  cos 2 x   tan x

sin x
9. cos x(tan x  cot x)  csc x
11.
csc x
 tan x  cot x
cos x
10. csc2 x(sec2 x  1) 
12. (sec x  tan x)2 
1
cos 2 x
1  sin x
1  sin x