- Welcome to a lesson on even and odd trigonometric identities. The goal of the video is to state and illustrate the even and odd trig identities. Let's start by talking about even functions. Even functions are symmetrical across the y-axis. Here are two functions that are even. If we were to fold these two functions across the y-axis, the right half would match up perfectly with the left half, meaning if we folded this across the y-axis, this half here would fall directly on this half here.

Another way to think of it is that every point on the right side of this function—let's say this point here—has a mirror image on the left side, this point right here. So we can say this function is a mirror image across the line y = x, and the same is true for the graph of y = cosine theta. If we were to fold this graph across the y-axis, it would match up perfectly with the other half.

This half here would match up perfectly with this half here. More specifically, if we take a look at this function here, f of 2 = 2. If we change the sign of the x coordinate, let's say we want f of -2, notice that f of -2 is also equal to +2. So if the function is even, if we change the sign of the x coordinate, the y coordinate or the function value remains the same.

To generalize this, we say f of x = f of -x if the function is even. For the trig function, if we consider x = pi/2, this point here, we know f of pi/2 = 0, and f of -pi/2 also = 0. So here are the even trig identities. Cosine of negative theta = cosine theta, and the same is true for the reciprocal function secant theta.

And again, you can see graphically that these are symmetrical across the y-axis. Now, let's talk about odd functions. Odd functions are symmetrical about the origin, which means they have rotational symmetry about the origin, which means if we rotate this 180 about the origin, the function would look exactly the same.

This blue half and this green half would just switch places if we rotated this 180 in either direction about the origin. And the same is true for the graph of y = sine theta. If we rotate this about the origin 180, this blue half and this green half would just switch places. So let's see what happens when we consider function values on odd functions.

Notice in this graph, f of 2 = +2, but f of -2, if we switch the sign of the x coordinate, notice how the y coordinate also switches signs. f of -2 = -2, so to state this relationship in general, we say that f of x = the opposite of f of -x, or if we multiply both of these by -1 and then flip it around, we can say that f of -x = the opposite of f of x, meaning if the function is odd and we change the sign of the x coordinates, the y coordinates or the function values will be the opposite sign.

For the trig function, notice that f of pi/2 = 1, and f of -pi/2 = -1. So let's take a look at the odd trig identities. Here they are for sine and cosecant, because these are odd functions, and the same is true for tangent and cotangent. Tangent and cotangent are odd functions. Therefore, they have rotational symmetry about the origin, and here's a summary of those identities.

Let's take a look at some examples. First example: if sine x = 0.75, then sine of -x is going to be = -0.75. The sine function is an odd function. So if we change the sign of the x coordinate, the function values will be the opposite sign. But the cosine function is an even function. So if we change the sign of the x coordinates on the sine function, the function values stay the same.

So if cosine x = 0.2, then cosine -x also = 0.2. The tangent function is an odd function. So if tangent x = 5.3, then tangent -x = -5.3. And our last example: the secant function is an even function. If secant -x = 2.9, then secant x = 2.9 as well. I hope you found these explanations helpful. Thank you for watching.