Trigonometric Identities

Pythagorean Identities



This is a Pythagorean Identity:

external image pythag5.gif


Using this first Pythagorean Identity, two additional Pythagorean Identities can be created.

The second Pythagorean Identity is:

external image pythag11.gif

external image pythag7.gif
Start with the first Pythagorean Identity.
Divide each term by external image pythag8.gif.

Remember:external image pythag10.gif
external image pythag9.gif
Reduce and Substitute.





The third Pythagorean Identity is:

external image pythag16.gif

external image pythag12.gif
Start with the first Pythagorean Identity.

Divide each term by external image pythag13.gif.

Remember:
external image pythag15.gifexternal image pythag14.gif
Reduce and Substitute.





How to Cancel Identities

If you have the opposite trigonometric functions, they cancel each other out to equal one.
tanx cotx= 1 ;because cotx=1/tanx then tanx times 1/tanx= tanx/tanx=1
cosx secx= 1
;because secx=1/cosx then cosx times 1/cosx= cosx/cosx=1
sinx cscx= 1;because cscx=1/sinx then sinx times 1/sinx= sinx/sinx=1


These three formulas can be algebraically altered in various ways to make other equations, as listed below:

  • sin²x + cos²x = 1 ---► 1 - sin²x = cos²x
  • sin²x + cos²x = 1 ---► 1 - cos²x = sin²x
  • tan²x + 1 = sec²x ---► sec²x - tan²x = 1
  • tan²x + 1 = sec²x ---► sec²x - 1 = tan²x
  • 1 + cot²x = csc²x ---► csc²x - cot²x = 1
  • 1 + cot²x = csc²x ---► csc²x - 1 = cot²x

Quotient Identities


tan θ =
sin θ
cos θ

cot θ =
cos θ
sin θ
This identity is used for changing tangent/cotangent into terms of sine and cosine, or vice-versa.

Reciprocal Identities


sin θ
=
1csc θ

csc θ
=
1 sin θ

cos θ
=
1sec θ

sec θ
=
1cos θ

tan θ
=
1cot θ

cot θ
=
1tan θ
When multiplying a trigonometric function by its inverse, the result is 1. For instance...
  • sin(θ) x csc(θ) = 1

As a numerical example, 1/2 x 2/1 = 1. In this case, the same principle applies.

This can also be used to move a denominator to the numerator (or vise-versa).
Ex: tan(x)/sec(x) can be changed to cos(x) tan(x).

Odd-Even Identities


sin(-x)= -sinx
cos(-x) = cosx
tan(-x) = -tanx
csc(-x)= -cscx
sec(-x)= secx
cot(-x)= -cotx
Odd functions have opposite end behavior.
Sine, cosecant, tangent, and co tangent are odd functions.

Even functions have the same end behavior.

Cosine and secant are even functions

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An Easy to Understand Lesson on Identities