An identity, in mathematics, is an equation that is true for all possible values.
(It is always true regardless of the values that are substituted into the equation.)
A trigonometric identity is an equation that involves trigonometric functions and is true for every single value substituted for the variable (assuming both sides are "defined" for that value) You will find that trigonometric identities are especially useful for simplifying trigonometric expressions.

The most common trigonometric identities are those involving the Pythagorean Theorem.


When studying the unit circle (radius of 1), it was observed that a point on the unit circle (a vertex of the right triangle) can be represented by the coordinates (cos θ, sin θ ). 

Since the legs of the right triangle in the unit circle have the values of sin θ and cos θ, the Pythagorean Theorem can be used to obtain sin2 θ + cos2 θ = 1.

pythag5
unitcircle2

This well-known equation is called a Pythagorean Identity. 
It is true for all values of θ in the unit circle. 

Using this first Pythagorean Identity, two additional Pythagorean Identities can be created.
pythagorean7
• Start with this first Pythagorean Identity.

• Divide each term by cos2θ.

• We know pyth8 and pyth0.


• Substitute and simplify.

We now have a second Pythagorean Identity:

pythag11

It should be noted that there are values of θ for which tangent and secant are undefined.
undefined


If we divide by a different value, we can arrive at the third identity:
pythagorean7
• Start with this first Pythagorean Identity.

• Divide each term by sin2θ.

• We know pyth14 and pyth15.


• Substitute and simplify.

The third Pythagorean Identity is:

pytha16
underined2


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When working with Pythagorean Identities, keep in mind that there are
other ways to express (rearrange) these identities that may be useful.
Pythagorean Identity Other Variation(s)
2 3              4
5                             6
7                             8



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ex1
Use Pythagorean Identities to find missing trigonometric values.
 
pyth24
find the value of the other trig functions.
Yes, this problem can also be solved without the use of a Pythagorean Identity.
 
Solution:
The easiest Pythagorean Identity to work with is
sin2θ +cos2θ = 1.
pyth25
pyth26
pythPicEx1
 
pyth27
 
pyth28
pyth29

 
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ex2
Use Pythagorean Identities to help simplify trig expressions.
 
Simplify:    sinx cos2x - sinx
  
 
Solution: Since this expression contains sine and cosine, utilize sin2θ +cos2θ = 1 or its variations.
Start by factoring:
pyth31
Use this substitution:
sin2x +cos2x = 1
gives
cos2x - 1 = -sin2x

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ex3
Use Pythagorean Identities to help simplify a trig expression into a factorable form.
 
Express csc2x - cotx - 3 in factored form.
  
 
Solution:
Start by substituting:
pyth31
Use this Pythagorean Identity:
1 + cot2x = csc2x
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You will find that these Pythagorean Identities
will also be valuable when solving trigonometric equations.

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ti84c
For help with verifying trig identities on
your calculator,
click here.
MathBits Calculator Pages


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