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I. Fundamental or Basic Identities
A. Reciprocal Identities
sin θ
csc θ =
cosθ
sec θ =
tanθ
cotθ =
B. Quotient Identities
cos θ
sin θ
tan θ =
sinθ
cos θ
cotθ =
C. Pythagorean Identities
sin θ+cos θ= 1
2 2
tan θ+ 1 =sec θ
2 2
cot θ+ 1 =csc θ
2 2
II. Identities for Negatives
sin(−θ) = − sinθ cos(−θ) = cosθ tan(−θ) = −tanθ
III. Co - function Identities
sin( 90 ±θ)=cos θ
0
sin(θ ± 90 )=±cosθ
0
cos( 90 ±θ)= sin θ
0
∓ cos(θ ± 90 )= sinθ
0
IV. Sum and Difference Identities
sin(α ±β)=sinαcosβ±cosαsin β
1 tanαtanβ
tanα±tan β
tan(α ±B) =
cos(α ±β)=cosαcosβ∓sinαsin β
V. Double - Angle Identities
sin 2 θ= 2 sinθcos θ
cos 2 θ = cos
2
θ − sin
2
θ = 1 − 2 sin
2
θ = 2 cos
2
θ − 1
1 tan θ
2tanθ
tan2θ
2
1 - cos2θ
sin θ
2
1 cos2θ
cos θ
2
VI. Half - Angle Identities
1 cos
sin
− θ
θ = ±
1 cos
cos
θ= ±
θ
− θ
θ
− θ
θ = ±
sin
1 cos
1 cos
sin
1 cos
1 cos
tan
Summary of Formula
Derivative of a Function by Formula / Rules for Differentiation
- The Constant Rule (c) 0
dx
d
- The Identity Function Rule (x) 1
dx
d
- The Constant & a Function Rule
dx
du
(cu) c
dx
d
4. The Sum / Difference Rule d( u+ v)=du+dv d( u−v)=du−dv
5. The Product Rule d( uv)=udv+vdu
- The Quotient Rule
2
v
vdu udv
v
u
d
7. The Power Rule / Formula d( u) nu du
n n− 1
- Chain Rule
dx
du
du
dy
dx
dy
- Inverse Function Rule
dy
dx
dx
dy
= , where 0
dy
dx
- Parametric Function Rule
du
dx
du
dy
dx
dy
= , where 0
du
dx
- Higher Parametric Function Rule
dx
dy
du
d
du
dx
dx
d y
2
2
Derivative of Trigonometric Functions
- ( )
dx
du
sinu cos u
dx
d
= 4. ( )
dx
du
secu secutan u
dx
d
- ( )
dx
du
cosu sin u
dx
d
= − 5. ( )
dx
du
cscu cscucot u
dx
d
- ( )
dx
du
tanu sec u
dx
d
2
= 6. ( )
dx
du
cotu csc u
dx
d
2
