Retest-10/22 pm or 10/23 am
Name___________________________________ ID: 1
Precalculus Trig Review
Verify each identity.
1) cot x + csc x =
1 + cos x
sin x
2) csc 2 x + cot 2 x =
cos 2 x + 1
sin 2 x
3) cot 2 x − tan 2 x = csc 2 x − sec 2 x
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4) 2sin xcos x + 1 + cos 2x = sin 2x + 2cos 2 x
5)
7)
cot 2 x − csc 2 x
= −cos xcsc x
tan x
cos x
sec 2 xcot x
=
sin x
1 + tan 2 x
6)
1
tan 2 x + 1
=
sin 2 x
tan 2 x
( )
8) tan θ +
tan θ + 1
π
=
4
1 − tan θ
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9) cos 2 xtan 2 x = cos 2 x − cos 2x
10) sin
(
)
3π
− θ = −cos θ
2
Solve each equation for 0 ≤ θ < 2π.
11) 3cos 2 θ = 7cos 2 θ − 3
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12) 0 = 2sin θ + 2 − cos 2 θ
13) 3sec 2 θ = −sec θ + 4sec 2 θ − 2
14) 2 3sec θ sin θ + 2sin θ = −3sec θ + 2sin θ
15) 3cot 2 θ = 1
16) −6 = cos θ − sin θ − 5
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Worksheet by Kuta Software LLC
Retest-10/22 pm or 10/23 am
Name___________________________________ ID: 1
Precalculus Trig Review
Verify each identity.
1) cot x + csc x =
1 + cos x
sin x
cot x + csc x
Decompose into sine and cosine
cos x
1
+
sin x sin x
Simplify
1 + cos x
sin x
■
2) csc 2 x + cot 2 x =
cos 2 x + 1
sin 2 x
Decompose into sine and cosine
csc 2 x + cot 2 x
( ) ( )
1
sin x
2
+
cos x
sin x
2
cos 2 x + 1
sin 2 x
Simplify
■
3) cot 2 x − tan 2 x = csc 2 x − sec 2 x
cot 2 x − tan 2 x
Use tan 2 x + 1 = sec 2 x
cot 2 x + 1 − sec 2 x
Use cot 2 x + 1 = csc 2 x
csc 2 x − sec 2 x
■
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4) 2sin xcos x + 1 + cos 2x = sin 2x + 2cos 2 x
5)
2sin xcos x + 1 + cos 2x
Use sin 2x = 2sin xcos x
sin 2x + 1 + cos 2x
Use cos 2x = 2cos 2 x − 1
sin 2x + 2cos 2 x
■
cot 2 x − csc 2 x
= −cos xcsc x
tan x
cot 2 x − csc 2 x
tan x
Use cot 2 x + 1 = csc 2 x
1
−
tan x
sin x
Use tan x =
cos x
cos x
−
sin x
1
Use csc x =
sin x
6)
tan 2 x + 1
1
=
sin 2 x
tan 2 x
tan 2 x + 1
1
sec x
sin 2 xcos 2 x
2
sin x
sec 2 xcot x
=
cos x
sec 2 xcot x
cos xsin x
sec 2 xcos x
sin x
sec 2 x
sin x
1 + tan 2 x
sin x
cos x
( )
sin x
8) tan θ +
1 + tan 2 x
Use cot x =
Use tan x =
■
tan 2 x
cos x
1
cos x
Create a common factor
cos 2 x
sin 2 x
7)
Use tan 2 x + 1 = sec 2 x
Use sec x =
2
■
−cos xcsc x
1
tan θ + 1
π
=
4
1 − tan θ
( )
cos x
sin x
tan θ +
π
4
π
4
=
π
1 − tan θ tan
4
tan θ + 1
=
1 − tan θ ⋅ 1
tan θ + 1
=
1 − tan θ
tan θ + tan
Cancel common factors
Use tan 2 x + 1 = sec 2 x
■
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9) cos 2 xtan 2 x = cos 2 x − cos 2x
Decompose into sine and cosine
cos 2 xtan 2 x
cos 2 x ⋅
( )
sin x
cos x
2
Simplify
sin 2 x
Use cos 2x = cos 2 x − sin 2 x
cos 2 x − cos 2x
■
10) sin
(
(
)
3π
− θ = −cos θ
2
)
3π
−θ
2
3π
3π
= sin
cos θ − cos
sin θ
2
2
= −cos θ − 0sin θ
= −cos θ
sin
Solve each equation for 0 ≤ θ < 2π.
11) 3cos 2 θ = 7cos 2 θ − 3
{
π 5π 7π 11π
,
,
,
6 6 6
6
}
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12) 0 = 2sin θ + 2 − cos 2 θ
13) 3sec 2 θ = −sec θ + 4sec 2 θ − 2
{ }
{
3π
2
π
5π
, π,
3
3
}
14) 2 3sec θ sin θ + 2sin θ = −3sec θ + 2sin θ
{
4π 5π
,
3 3
}
16) −6 = cos θ − sin θ − 5
15) 3cot 2 θ = 1
{
π 2π 4π 5π
,
,
,
3 3 3 3
{ }
}
π
,π
2
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