Calculus Trig and Hyper Identities

sin(A+B) (Sum/Difference Identities)

sinAcosB+cosAsinB

sin(A−B) (Sum/Difference Identities)

sinAcosB−cosAsinB

cos(A+B) (Sum/Difference Identities)

cosAcosB−sinAsinB

cos(A−B) (Sum/Difference Identities)

cosAcosB+sinAsinB

sin(−θ) (Even/Odd Identities)

−sinθ

cos(−θ) (Even/Odd Identities)

cosθ

tan(−θ) (Even/Odd Identities)

−tanθ

csc(−θ) (Even/Odd Identities)

−cscθ

sec(−θ) (Even/Odd Identities)

secθ

cot(−θ) (Even/Odd Identities)

−cotθ

sin(2θ) (Double Angle Identities)

2sinθcosθ

cos(2θ) (Double Angle Identities)

1. cos²(θ) − sin²(θ)

2. 1 − 2sin²(θ)

3. 2cos²(𝜃) - 1

sin²(θ) (Power-Reducing Identities)

(1 - cos2θ) / 2

cos²(θ) (Power-Reducing Identities)

(1 + cos2θ) / 2

sin(π/2 ​− θ) (Cofunction Identities)

cosθ

cos(π/2 ​− θ) (Cofunction Identities)

sinθ

d/dx(tanx) (Trigonometric Derivatives)

sec²(x)

d/dx(cotx) (Trigonometric Derivatives)

−csc²(x)

d/dx(secx) (Trigonometric Derivatives)

sec(x)tan(x)

d/dx(cscx) (Trigonometric Derivatives)

−csc(x)cot(x)

Difference of cubes

a³ - b³ = (a - b)(a² + ab + b²)

Sum of Cubes

a³ + b³ = (a + b)(a² - ab + b²)

Sinh(x) Definition

( e^x - e^-x ) / 2

Cosh(x) Definition

( e^x + e^-x ) / 2

Hyperbolic Fundamental Identity

cosh²(x) - sinh²(x) = 1

cosh²(x)

( cosh(2x) + 1) / 2

sinh²(x)

( cosh(2x) - 1) / 2

sinh⁻¹(x)

ln ( x + √x² + 1)

cosh⁻¹(x)

ln ( x + √x² - 1)

sinh⁻¹(u) + C

∫ du / √(1 + u²)

cosh⁻¹(u) + C

∫ du / √( u² - 1 )

tanh⁻¹(u) + C

∫ du / ( 1 - u²)

sin⁻¹(u) + C

∫ du / √( 1 - u² )

tan⁻¹(u) + C

∫ du / ( 1 + u² )

Sin(A)Cos(B) (Product-to-sum)

½ [sin(A + B) + sin(A - B)]

lim (n→∞) [ ln(n) / n]

0

lim (n→∞) [ x^(1/n) ]

1 ( x > 0) )

lim (n→∞) [ ( 1 + x/n )^n ]

e^x

lim (n→∞) [ n√ (n) ]

1

lim (n→∞) [ x^n ]

0 ( |x| < 1)

lim (n→∞) [ x^n / n! ]

0

Indeterminate forms that needed solving or rearrange for L’H

1. [(0)(∞)]

2. [∞−∞]

3. [1^∞]

4. [0^∞]

5. [∞^0]

Results for Geometric Series

|r| < 1 → Converges

|r| ≥ 1 → Diverges

Results for P series

p > 1 → Converges

p ≤ 1 → Diverges

Results for Ratio & Root Test

p < 1 → Converge

p > 1 → Diverge

p = 1 → Fails