Pythagorean Identities

Pythagorean Identities in trigonometry will show up very frequently and can be very useful.  I will explain how Pythagorean Identities get their name, how you can derive them, and how you can remember them.  First, it would be a good idea for you to be able to understand the basic trig functions sine, cosine, and tangent.  Once you are familiar with these trig equations, the algebra that we will apply to them will allow us to derive the Pythagorean Identities. I have prepared other posts on this site that are dedicated to sine, cosine, and tangent that may be useful to review.

The Pythagorean Identities get their name because they are based on the famous Theorem of Pythagoras.  You are very likely already familiar with it. (On a side note, here are some interesting facts about the Theorem of Pythagoras.) Simply, for a right angle triangle, it says “the square of the hypotenuse is the sum of the squares of the other two sides.”  Mathematically, you have seen this represented as:

a2 + b2 = c2, where a and b are sides and c is the hypotenuse. 

Now, I will show you how to derive these special trig identities, using this theorem as our starting point.  To do this, we need to start with a right triangle, created by the radius of a unit circle and the axis:

unit circle to derive Pythagorean Identities


We can say that the right triangle formed by dropping a line from the point that the radius touches the circle (anywhere in quadrant I is sufficient for this demonstration) down to the axis has a base of x units long and y units high. (The actual numbers are not important, but they will depend on the specific angle, if you did need to calculate them for whatever reason. You don’t here.)  The radius in a unit circle, by definition, is 1.  Now, let’s apply the definitions of sine and cosine to our triangle.  Recall:

sin(ɵ) = opposite / hypotenuse = y / 1 = y

cos(ɵ) = adjacent / hypotenuse = x / 1 = x

So, now we can relabel our diagram by substituting in these basic trig identities.

unit circle to derive Pythagorean Identities 2
With the triangle now correctly labeled for our derivation, we can apply the Theorem of Pythagoras to arrive at one of the Pythagorean Identities. Since a2 + b2 = c2, we can therefore equate the sides of our triangle to these terms to give us our first of the trig Pythagorean Identities:

sin2(ɵ) + cos2(ɵ) = 1

If you have followed along up till now and understood everything I’ve done, then you are well on your way to remembering this trigonometric identity.  If you can remember how to derive it, you don’t even have to memorize it (though it always helps!)  For the next Pythagorean Identity, you start with this first identity, and you apply some basic algebra and trigonometry to it to derive the second and third identities.  Recall the definitions of secant, cosecant, and cotangent:

sec(ɵ) = 1 / cos(ɵ)

csc(ɵ) = 1 / sin(ɵ)

cot(ɵ) = 1 / tan(ɵ) = cos(ɵ) / sin(ɵ)

With those inverse trig functions in mind, let’s take the first Pythagorean Identity and divide all of its terms by cos2(ɵ).  That gives you: 

1 / cos2(ɵ) = sin2(ɵ) / cos2(ɵ) + cos2(ɵ) / cos2(ɵ)

sec2(ɵ) = tan2(ɵ) + 1

And this is the second Pythagorean Identity!  Using the same strategy we just used to derive that one, go back to the first one and divide everything by sin2(ɵ), to arrive at the third Pythagorean Identity!

csc2(ɵ) = 1 + cot2(ɵ)

I hope that from this tutorial, you now understand how these identities get their name, how you can derive them, and how to use this knowledge to help you to memorize or recall them.  Using the fundamental trigonometry identities and trig relations, it is easy to come up with more advanced trigonometric formulas.  If you need to refer back to this Pythagorean Identities list, please bookmark this page and come back again.  Also, be sure to follow me on Facebook/Twitter/Google+ (@TheNumerist).  Thanks.

(This is an old post from my previous math site, In Mathematics, copied here to consolidate all my math pages.)

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