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- Fearless Trigonometry – The Pythagorean Identities
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## Fearless Trigonometry – The Pythagorean Identities

The famous Pythagorean theorem is extended to trigonometry using the Pythagorean identities. Of course, the Pythagorean theorem is best remembered for the equation a^2 + b^2 = c^2. To extend this to trigonometry, let (x, y) be an ordered pair on the unit circle, that is, the circle centered at the origin and of radius 1. By our famous theorem, we have that x^2 + y ^2 = 1, since the xiy coordinates form a right triangle with hypotenuse 1. It is from this construction that we obtain the trigonometric identities, which we explore here.

Let’s remember the definitions of the sine and cosine functions in the unit circle of the equation x^2 + y^2 = 1. To understand it, it is important to know that the x-coordinate is the abscissa and the y-coordinate is the orderly With this in mind, we define sine as the ordinate/radius and cosine as the abscissa/radius. Denoting xiy as the abscissa and ordinate, respectively, ir as the radius, and A as the generated angle, we have sin(A) = y/ri cos(A) = x/r.

Since r = 1, sin(A) = yi cos(A) = x in the previous definitions. Since we know that x^2 + y^2 = 1, we have sin^2(A) + cos^2(A) = 1. This is our first Pythagorean identity based on the unit circle. Now there are two more based on the other trigonometric functions, namely the tangent, cotangent, secant, and cosecant. Fortunately, you only need to memorize the first one because the other two are free, as my Calculus I teacher taught me during my freshman year in college. The way to derive the other two identities is based on the relationship between tangent (tan) and cotangent (cot); and secant (sec) and cosecant (csc).

**Reciprocal identities**

To derive the other two Pythagorean identities, we use the following reciprocal identities:

**csc(A) = 1/sin(A)**

**sec(A) = 1/cos(A)**

**cot(A) = 1/tan(A)**

**tan(A) = sin(A)/cos(A)**

As my college calculus teacher showed me, we start with the first one and successively get the others as follows:

**(1) sin^2(A) + cos^2(A) = 1**

To get the Pythagorean identity that includes tan and cot, we divide the entire equation by cos^2(A). This gives

**sin^2(A)/cos^2(A) + cos^2(A)/cos^2(A) = 1/cos^2(A)**

Using the reciprocal identities above, we see that this equation is the same as

**tan^2(A) + 1 = sec^2(A)**

To obtain the Pythagorean identity involving cot and csc, we divide equation (1) above by sin^2(A), returning to our reciprocal identities to obtain

**sin^2(A)/sin^2(A) + cos^2(A)/sin^2(A) = 1/sin^2(A)**

Simplifying, this gives our third Pythagorean identity:

**1 + cot^2(A) = csc^2(A)**

That’s really all there is to it. And that my dear friends is how we use one identity to get two for free. There may be no free lunches in life, but at least sometimes there are free lunches in math. Thank God!

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