Reciprocal Identities Co-function identities Double Angle Form
1 𝜋 sin(2𝑢) = 2 sin 𝑢 co
sin 𝑥 = sin( − 𝑥) = cos 𝑥
csc 𝑥 2
cos(2𝑢) = cos2 𝑢 −
1 𝜋
tan 𝑥 = tan( − 𝑥) = cot 𝑥 2 tan 𝑢
cot 𝑥 2 tan(2𝑢) =
𝜋 1 − tan2
1 csc( − 𝑥) = sec 𝑥
cos 𝑥 = 2
sec 𝑥
*same for the inverse Half Angle Formul
*same for the inverse
1 − cos(2𝑢
Parity Identities sin2 𝑢 =
Pythagorean Identities 2
sin(−𝑥) = − sin 𝑥 1 + cos(2𝑢
sin2 𝑥 + cos2 𝑥 = 1
cos2 𝑢 =
tan(−𝑥) = − tan 𝑥 2
1 + tan2 𝑥 = sec 2 𝑥
cos(−𝑥) = cos 𝑥 1 − cos(2𝑢
1 + cot 2 𝑥 = csc 2 𝑥 tan2 𝑢 =
*same for the reciprocal 1 + cos(2𝑢
1 − cos(𝑢)
Ratio Identities tan2 𝑢 =
Sum & Difference Formula sin 𝑢
sin 𝑥 sin 𝑢
tan 𝑥 = sin(𝑢 ± 𝑣) = sin 𝑢 cos 𝑣 ± cos 𝑢 sin 𝑣 tan2 𝑢 =
cos 𝑥 1 + cos(𝑢)
cos 𝑥 cos(𝑢 ∓ 𝑣) = cos 𝑢 cos 𝑣 ± sin 𝑢 sin 𝑣
cot 𝑥 =
sin 𝑥 tan 𝑢 ± tan 𝑣
tan(𝑢 ± 𝑣) =
1 ∓ tan 𝑢 tan 𝑣
, Sum to Product Formula 𝑒 𝑥𝑙𝑛𝑎 = 𝑎 𝑥 Fundamental Rule
𝑢+𝑣 𝑢−𝑣 ln 1 = 0 𝐶′ = 0
sin 𝑢 + sin 𝑣 = 2 sin ( ) cos ( )
2 2
𝑢+𝑣 𝑢−𝑣 𝑒0 = 1 (𝑢 + 𝑣)′ = 𝑢′ + 𝑣′
sin 𝑢 − sin 𝑣 = 2 cos ( ) sin ( ) 1
2 2 =∞ (𝑢𝑣)′ = 𝑢𝑣′ + 𝑣𝑢′
0
𝑢+𝑣 𝑢−𝑣
cos 𝑢 + cos 𝑣 = 2 cos ( ) cos ( ) 1 𝑢 𝑣𝑢′ − 𝑢𝑣′
2 2 =0 ( )′ =
∞ 𝑣 𝑣2
𝑢+𝑣 𝑢−𝑣
cos 𝑢 + cos 𝑣 = −2 sin ( ) sin ( ) ln( −𝑎) = ∞ (𝑢𝑛 )′ = 𝑛(𝑢𝑛−1 )𝑢′
2 2
log( −𝑎) = ∞
Product to Sum Formula
𝑎 √𝑎 Logarithms and E
1 √𝑏 =
√𝑏
sin 𝑢 sin 𝑣 = [cos(𝑢 − 𝑣) − cos(𝑢 + 𝑣)] 𝑢′
2 ln 𝑢 =
1 √𝑎𝑏 = √𝑎√𝑏 𝑢
cos 𝑢 cos 𝑣 = [cos(𝑢 − 𝑣) + cos(𝑢 + 𝑣)]
2 ln 𝑎 + ln 𝑏 = (ln 𝑎) ln 𝑏 𝑒 𝑢 = 𝑒 𝑢 𝑢′
1 𝑢′
sin 𝑢 cos 𝑣 = [sin(𝑢 + 𝑣) + sin(𝑢 − 𝑣)] ln 𝑎
2 ln 𝑎 − ln 𝑏 = log 𝑎 𝑢 =
ln 𝑏 (ln 𝑎)𝑢
1
cos 𝑢 sin 𝑣 = [sin(𝑢 + 𝑣) − sin(𝑢 − 𝑣)] 𝑒 𝑎+𝑏 = 𝑒 𝑎 (𝑒 𝑏 ) log 𝑒 𝑢′
2
log 𝑎 𝑢 =
𝑢
Properties to Remember Derivatives 𝑎𝑢 = (ln 𝑎)𝑎𝑢 𝑢′
𝑎𝑢 = 𝑒 𝑢 ln 𝑎 (𝑢 ln 𝑎)′
1 𝜋 sin(2𝑢) = 2 sin 𝑢 co
sin 𝑥 = sin( − 𝑥) = cos 𝑥
csc 𝑥 2
cos(2𝑢) = cos2 𝑢 −
1 𝜋
tan 𝑥 = tan( − 𝑥) = cot 𝑥 2 tan 𝑢
cot 𝑥 2 tan(2𝑢) =
𝜋 1 − tan2
1 csc( − 𝑥) = sec 𝑥
cos 𝑥 = 2
sec 𝑥
*same for the inverse Half Angle Formul
*same for the inverse
1 − cos(2𝑢
Parity Identities sin2 𝑢 =
Pythagorean Identities 2
sin(−𝑥) = − sin 𝑥 1 + cos(2𝑢
sin2 𝑥 + cos2 𝑥 = 1
cos2 𝑢 =
tan(−𝑥) = − tan 𝑥 2
1 + tan2 𝑥 = sec 2 𝑥
cos(−𝑥) = cos 𝑥 1 − cos(2𝑢
1 + cot 2 𝑥 = csc 2 𝑥 tan2 𝑢 =
*same for the reciprocal 1 + cos(2𝑢
1 − cos(𝑢)
Ratio Identities tan2 𝑢 =
Sum & Difference Formula sin 𝑢
sin 𝑥 sin 𝑢
tan 𝑥 = sin(𝑢 ± 𝑣) = sin 𝑢 cos 𝑣 ± cos 𝑢 sin 𝑣 tan2 𝑢 =
cos 𝑥 1 + cos(𝑢)
cos 𝑥 cos(𝑢 ∓ 𝑣) = cos 𝑢 cos 𝑣 ± sin 𝑢 sin 𝑣
cot 𝑥 =
sin 𝑥 tan 𝑢 ± tan 𝑣
tan(𝑢 ± 𝑣) =
1 ∓ tan 𝑢 tan 𝑣
, Sum to Product Formula 𝑒 𝑥𝑙𝑛𝑎 = 𝑎 𝑥 Fundamental Rule
𝑢+𝑣 𝑢−𝑣 ln 1 = 0 𝐶′ = 0
sin 𝑢 + sin 𝑣 = 2 sin ( ) cos ( )
2 2
𝑢+𝑣 𝑢−𝑣 𝑒0 = 1 (𝑢 + 𝑣)′ = 𝑢′ + 𝑣′
sin 𝑢 − sin 𝑣 = 2 cos ( ) sin ( ) 1
2 2 =∞ (𝑢𝑣)′ = 𝑢𝑣′ + 𝑣𝑢′
0
𝑢+𝑣 𝑢−𝑣
cos 𝑢 + cos 𝑣 = 2 cos ( ) cos ( ) 1 𝑢 𝑣𝑢′ − 𝑢𝑣′
2 2 =0 ( )′ =
∞ 𝑣 𝑣2
𝑢+𝑣 𝑢−𝑣
cos 𝑢 + cos 𝑣 = −2 sin ( ) sin ( ) ln( −𝑎) = ∞ (𝑢𝑛 )′ = 𝑛(𝑢𝑛−1 )𝑢′
2 2
log( −𝑎) = ∞
Product to Sum Formula
𝑎 √𝑎 Logarithms and E
1 √𝑏 =
√𝑏
sin 𝑢 sin 𝑣 = [cos(𝑢 − 𝑣) − cos(𝑢 + 𝑣)] 𝑢′
2 ln 𝑢 =
1 √𝑎𝑏 = √𝑎√𝑏 𝑢
cos 𝑢 cos 𝑣 = [cos(𝑢 − 𝑣) + cos(𝑢 + 𝑣)]
2 ln 𝑎 + ln 𝑏 = (ln 𝑎) ln 𝑏 𝑒 𝑢 = 𝑒 𝑢 𝑢′
1 𝑢′
sin 𝑢 cos 𝑣 = [sin(𝑢 + 𝑣) + sin(𝑢 − 𝑣)] ln 𝑎
2 ln 𝑎 − ln 𝑏 = log 𝑎 𝑢 =
ln 𝑏 (ln 𝑎)𝑢
1
cos 𝑢 sin 𝑣 = [sin(𝑢 + 𝑣) − sin(𝑢 − 𝑣)] 𝑒 𝑎+𝑏 = 𝑒 𝑎 (𝑒 𝑏 ) log 𝑒 𝑢′
2
log 𝑎 𝑢 =
𝑢
Properties to Remember Derivatives 𝑎𝑢 = (ln 𝑎)𝑎𝑢 𝑢′
𝑎𝑢 = 𝑒 𝑢 ln 𝑎 (𝑢 ln 𝑎)′
