Trigonometric Identities sec x = 1 cos x csc x = 1 sin x tan x = sin x

Trigonometric Identities
1
cos x
1
csc x =
sin x
sin x
tan x =
cos x
cos x
cot x =
sin x
sin2 x + cos2 x = 1
sec x =
sec2 x = 1 + tan2 x
csc2 x = 1 + cot2 x
sin(x ± y) = sin x cos y ± cos x sin y
cos(x ± y) = cos x cos y ∓ sin x sin y
1
1
sin x sin y = cos(x − y) − cos(x + y)
2
2
1
1
cos x cos y = cos(x − y) + cos(x + y)
2
2
1
1
sin x cos y = sin(x − y) + sin(x + y)
2
2
sin 2x = 2 sin x cos x
cos 2x = cos2 x − sin2 x
cos 2x = 1 − 2 sin2 x
cos 2x = 2 cos2 x − 1
1 − cos 2x
sin2 x =
2
1 + cos 2x
2
cos x =
2
Derivatives
d
[f (x)g(x)] = f (x)g 0 (x) + g(x)f 0 (x)
dx
d f (x)
g(x)f 0 (x) − f (x)g 0 (x)
=
dx g(x)
[g(x)]2
d
[f (g(x))] = f 0 (g(x))g 0 (x)
dx
d n
[x ] = nxn−1
dx
d x
[e ] = ex
dx
d x
[a ] = ax ln a
dx
1
d
[ln x] =
dx
x
d
[sin x] = cos x
dx
d
[cos x] = − sin x
dx
d
[tan x] = sec2 x
dx
d
[csc x] = − csc x cot x
dx
d
[sec x] = sec x tan x
dx
d
[cot x] = − csc2 x
dx
d
1
[sin−1 x] = √
dx
1 − x2
d
1
[tan−1 x] = 2
dx
x +1
−1
d
[cos−1 x] = √
dx
1 − x2
−1
d
[cot−1 x] = 2
dx
x +1
d
1
√
[sec−1 x] =
dx
|x| x2 − 1
d
−1
√
[csc−1 x] =
dx
|x| x2 − 1
Integrals
Z
xn+1
+ C if n 6= −1
xn dx =
n+1
Z
1
dx = ln |x| + C
x
Z
ex dx = ex + C
Z
ax
ax dx =
+C
ln a
Z
sin x dx = − cos x + C
Z
cos x dx = sin x + C
Z
sec2 x dx = tan x + C
Z
csc2 x dx = − cot x + C
Z
sec x tan x dx = sec x + C
Z
csc x cot x dx = − csc x + C
Z
tan x dx = − ln | cos x| + C = ln | sec x| + C
Z
cot x dx = ln | sin x| + C
Z
sec x dx = ln | sec x + tan x| + C
Z
Z
Z
Z
Z
Z
Z
csc x dx = ln | csc x − cot x| + C
Z
u dv = uv − v du
1
x
dx = sin−1 + C
a
a2 − x2
1
1
x
dx = tan−1 + C
x2 + a2
a
a
x
1
1
√
dx = sec−1 + C
a
a
x x2 − a 2
Z
n−1
1
sinn−2 x dx
sinn x dx = − sinn−1 x cos x +
n
n
Z
1
n−1
n
n−1
cos x dx = cos
x sin x +
cosn−2 x dx
n
n
√
Power Series
sin x =
∞
X
(−1)n 2n+1
x
(2n + 1)!
n=0
cos x =
∞
X
(−1)n 2n
x
(2n)!
n=0
ex =
∞
X
1 n
x
n!
n=0
ln(1 + x) =
IOC = (−∞, ∞)
IOC = (−∞, ∞)
IOC = (−∞, ∞)
∞
X
(−1)n−1 n
x
n
n=1
IOC = (−1, 1]
∞
X
1
=
xn
1 − x n=0
tan−1 x =
IOC = (−1, 1)
∞
X
(−1)n 2n+1
x
2n + 1
n=0
IOC = [−1, 1]