Derivative of cosec(x)

Introduction

In this article, we are going to discuss the visual derivation of the Derivative of cosec(x). cosec(x) is defined as one over sin(x) or in general it is the ratio of the hypotenuse and the opposite side to the angle, in a right angled triangle.

Consider a unit circle where the hypotenuse subtends an angle of x radians. The length of the adjacent side AB in this scenario corresponds to the value of sin(x).

Small change in angle

Now, we will increase the angle by a small value dx as shown above, this will increase the length of the opposite side, the length of the new side will be sin(x+dx) represented by CD.

Note: Before proceeding further with this article, I would highly recommend you to go through the articles explaining the derivatives of sin(x) and cos(x), if you have not already gone through them. They would give you the prerequisite understanding required to proceed further with this proof.

Change in sin(x)

From the above equation we can observe that , when we increase the angle x by a small value dx the sin(x) side will increase by a length of cos(x) times dx.

Remember that dx is a very small value and cos(x) dx are also very small values. Now, using these equations we have to compare cosec(x) and cosec(x+dx), to get the change in the function cosec(x).

Meaning of cosec(x)

To visualize the meaning of cosec(x) we have taken some appropriate values of Hypotenuse(which is one in our case) and sin(x) as lengths of lines to visualize the process of obtaining cosec(x). We can observe in the above figure that cosec(x) is nothing but the number of parts of sin(x) that can fit completely inside Hypotenuse. Along with whole parts if there is some residual length, we can also use fractional parts of sin(x) to completely fill the Hypotenuse. Therefore, cosec(x) is just the number of parts of sin(x) that can be placed inside Hypotenuse.

Change in cosec(x)

The idea is to figure out how many sin(x+dx) parts can fit into hypotenuse side. In the above image you can see the length of sin(x) shown in red and the additional length shown in blue. The first thing we want to do is to arrange an equal number of sin(x+dx) parts as compared to the previous sin(x) parts in the Hypotenuse side. We do this to get the same cosec(x) value and from there we can see how much the value increases or decreases.

Next, we can observe that since the individual sin(x+dx) parts are bigger, the combined length of the parts surpasses the length of the hypotenuse side, but we don’t want this, so we need to figure out how much of these parts we need to remove. Here, we can see that the total additional length is solely due to the change in sin(x) length which is cos(x) dx times the number of parts put together.

In the above equation, the first part represents how many times the sin(x) part is inserted in the hypotenuse line, the blue regions are the additional length in each part and we can observe that the number of those parts will be equal to the number of red parts, hence the second part of the equation, and each blue region has a length of cos(x) dx, therefore by multiplying the second and third parts of the equation we get the total length of the blue region which is the excess length, hence by dividing this with the denominator sin(x) we get the total value of the part that needs to be removed to make the sin(x+dx) parts to fit properly inside hypotenuse side, and hence giving us the left hand side of the equation.

Simplification

Now, bringing all the parts of the equations together and simplifying them will give us the change in the cosec(x) function.

Derivative of cosec(x)

Finally, taking the ratio of the functional change of cosec(x) and the change in the angle dx , we can arrive at the derivative of cosec(x).