Math Identities Proofs


Math Identities Proofs through Derivatives 2023 (PDF)

Math Identities Proofs through Trigonometry 2023 (PDF)

Math Identities Proofs through Algebra 2023 (PDF)

  • Ordered and proven by category without skipping i.e. assumed as fact
  • Internally hyperlinked navigation for easy reference
  • Variables confined to x, y, z, θ, etc. for easy understanding
  • Excellent supplemental resource for classes through Calculus I
  • Mathematical Basics
    • Arithmetic
    • Number Types
    • Functions
    • Complex Numbers
  • Exponentiation
    • Exponents
    • Polynomials
    • Logarithms
  • Basic & Analytic 2D Geometry
    • Lines
    • Triangles
    • Polygons
    • Conic Sections
  • Trigonometry
    • Definitions & Basics
    • Complex Unit Circle
    • Common Identities
    • Polar ↔ Rectangular Conversion
  • Limits and Derivatives
    • Limits Definition and Properties
    • Limits Common Rules
    • Mathematical Constants
    • Differentiation
    • Derivatives Common Rules
    • Trigonometric Derivatives
    • Complex Analysis and Trigonometry
Future updates for existing sections:
  • Conic sections x·y term solving
  • Parametric graphing in the trig section
  • The plastic ratio proof in constants
  • hyperbolic trig functions in derivatives
Future sections:
  • Series and integration of one variable (C)
  • Matrices and vectors (C)
  • Basic & analytic 3D geometry (R)
  • Multivariable calculus (C)
  • Differential geometry (C)

Other Resources

Paul's Online Notes
Be sure to check out it if you are struggling to conceptualize problems or need help with homework. This has proven to be an effectual tutoring resource. His website covers everything from the basics to differential equations with step-by-step solutions and cheat sheets in each subject.

Wolfram Alpha
An online calculator service in an encyclopedic-type format when yielding results. Inputs work similarly to inputting into TI series calculators, only it is much more powerful. There is a $5/month paid version which allows for viewing step-by-step solutions. I highly recommend this once at the calculus level.

I started working on this project back in 2013 for two reasons. One was reimagining the hundreds of flash cards I had created while in calculus classes. The other was because I had noticed certain websites and even textbooks would be wrong from time to time, giving conflicting or weird information. Math is an absolute construct that needs to be correct. With limited access to MS word, it sat with time passing it by. By 2016 I had done all the work I was going to do on the original version.

In 2020, I picked it up again briefly, but this time reformatting it to make it look better as well as going back over to make sure it was completed section by section before moving on to the next. It still pretty much sat though until Nov 2022. So the contents of the document now is actually lower tier, yet much more inclusive than what I already have in the original version. I wanted to ensure that things were proven before use in further proofs so that there is a visible logical progression as seen in the internal hyperlinks. This, as opposed to slopped together with circular arguments as seen on internet sites.

There are several different proofs for some mathematical equations, for example, e^(i*x). However, not all of those proofs are on a lower level than handling this expression. The earliest is in derivatives. There are proofs involving series expansion (the original), derivatives in polar coordinates, in the natural logarithm which sounds backwards, I'm sure there's more. This begs the question, what if you are in basic calculus and haven't gotten to any of that yet, and still want to understand where it came from? It is difficult to interpret where it came from when in most calc classes this is one of the first derivatives they teach, and only presenting it as a fact without any real context. Enter the structure of Math Identities Proofs.

Updated July 2023
Bill Liam East
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