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.

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


An Easy to Understand Lesson on Identities