Derivation of the Pythagorean Trigonometric Identities
Pronunciation: /ˌdɛɹ.ɪˈveɪ.ʃən ʌv ðə pɪˌθæ.gəˈri.ən ˌtrɪɡ.ə.nəˈmɛ.trɪk aɪˈdɛn.tə.tiz/ Explain
Derivation of
Image  Equation  Discussion 

A^{2} + B^{2} = C^{2} 
This is the equation given by the Pythagorean Theorem. 


Use the multiplicative property of equality to multiply both sides by . 


Since anything except zero divided by itself is one, substitute 1 for . 


Use the distributive property of multiplication over addition and subtraction to distribute through the parentheses on the left side of the equation. 


Apply the distributive property of exponentiation over division. 


The definition of sine is . The definition of cosine is . Given angle θ, and . Apply the substitution property of equality to substitute sin θ for and cos θ for . 
Derivation of
Image  Equation  Discussion 


Start with the first Pythagorean identity. 


Apply the multiplication property of equality to the equation by multiplying both sides of the equation by . 


Apply the distributive property of multiplication over addition and subtraction. 


Since anything except zero divided by itself is one, substitute 1 for . 


Since 1^{2} = 1, apply the substiution property of equality to subtitute 1^{2} = 1 for 1. 


Apply the distributive property of exponentiation over division to both sides of the equation. 


Use the definition of tangent to subsitute tan θ for . 


Use the definition of secant to subsitute sec θ for . 
Derivation of
Image  Equation  Discussion 


Start with the first Pythagorean identity. 


Apply the multiplication property of equality to the equation by multiplying both sides of the equation by . 


Apply the distributive property of multiplication over addition and subtraction. 


Since anything except zero divided by itself is one, substitute 1 for . 


Apply the commutative property of addition. 


Since 1^{2} = 1, apply the substiution property of equality to subtitute 1^{2} = 1 for 1. 


Apply the distributive property of exponentiation over division to both sides of the equation. 


Use the definition of tangent to subsitute cot θ for . 


Use the definition of secant to subsitute csc θ for . 
References
 McAdams, David E.. All Math Words Dictionary, derivation. 2nd Classroom edition 201501084799968. pg 58. Life is a Story Problem LLC. January 8, 2015. Buy the book
Cite this article as:
McAdams, David E. Derivation of the Pythagorean Trigonometric Identities. 3/10/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/d/dti_pythagorean.html.
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Revision History
12/21/2018: Reviewed and corrected IPA pronunication. (
McAdams, David E.)
7/4/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (
McAdams, David E.)
4/29/2011: Initial version. (
McAdams, David E.)