Derived Pythagorean Identities
One identity becomes three. By dividing by or , you unlock two powerful new identities involving tangent, secant, cotangent, and cosecant.
Introduction
The primary Pythagorean identity is a relationship between sine and cosine. But what about tangent and secant, or cotangent and cosecant? By performing a single algebraic step — dividing every term — you can derive two additional identities that are just as important.
Past Knowledge
and the quotient/reciprocal identities from 5.2.
Today's Goal
Derive and apply and .
Future Success
These identities appear heavily in Lessons 5.7–5.9 and in solving trig equations in Chapter 6.
Key Concepts
Derivation 1: Divide by
Start with and divide every term by :
Derivation 2: Divide by
Divide every term of the primary identity by :
The Complete Pythagorean Family
| # | Identity | Derived By |
|---|---|---|
| 1 | Unit circle (primary) | |
| 2 | Dividing #1 by | |
| 3 | Dividing #1 by |
Memory Tip
Each identity pairs a “regular” function with its reciprocal: sin↔csc, cos↔sec, tan↔cot. The squared reciprocal is always on the right side, and the is always present.
Worked Examples
Finding Secant from Tangent
Question: If and is in Quadrant III, find .
Step 1: Use :
Step 2: Take the square root. In QIII, cosine is negative, so secant is also negative:
Final Answer:
Simplifying an Expression
Question: Simplify .
Step 1: Recognize the rearranged form of the second Pythagorean identity:
Final Answer:
Factoring with Pythagorean Identities
Question: Simplify .
Step 1: Factor as a difference of squares:
Step 2: From , we know :
Step 3: Replace with :
Final Answer:
Common Pitfalls
Swapping the Identities
It's , not . The squared reciprocal function is always the larger value (since adding 1 makes it bigger). Secant is always ≥ 1 in absolute value, so it belongs on the big side.
Real-Life Applications
Structural Engineering — Inclined Loads
When engineers analyze forces on a sloped surface (like a ramp, roof, or bridge), they decompose forces into components using tangent and secant. The identity is used to relate the slope angle to the total force magnitude, simplifying equilibrium calculations for safe structural design.
Practice Quiz
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