# In this article, we will explore how to find the derivative of the product of two trigonometric functions, sin(x) and cos(x), and discover that the answer is indeed cos2(x). Calculus is a fundamental branch of mathematics that deals with the concepts of change and motion. It provides powerful tools for understanding how functions behave, and one of the key operations in calculus is finding derivatives. Derivatives measure the rate at which a function changes with respect to its input, and they have numerous applications in science, engineering, and everyday life.

### Understanding Trigonometric Functions

Before we dive into finding the derivative of sin(x) * cos(x), it's essential to have a solid grasp of the trigonometric functions involved.

#### Sine Function (sin(x)):

The sine function, denoted as sin(x), represents the ratio of the length of the side opposite an angle in a right triangle to the length of the hypotenuse. In the context of the unit circle, sin(x) is the y-coordinate of a point on the circle when the angle x is measured from the positive x-axis in a counterclockwise direction. The sine function oscillates between -1 and 1 as x varies.

#### Cosine Function (cos(x)):

The cosine function, denoted as cos(x), represents the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle. Similarly, in the unit circle context, cos(x) is the x-coordinate of a point on the unit circle when the angle x is measured from the positive x-axis in a counterclockwise direction. The cosine function also oscillates between -1 and 1 as x varies.

### Finding the Derivative of sin(x) * cos(x)

To find the derivative of sin(x) * cos(x), we can use the product rule, one of the fundamental rules of differentiation in calculus. The product rule states that if you have two functions, u(x) and v(x), then the derivative of their product is given by: d(uv)/dx = u'v + uv'

In our case, u(x) is sin(x), and v(x) is cos(x). We'll start by finding the derivatives of these functions:

#### Derivative of sin(x):

The derivative of sin(x) with respect to x is cos(x). This result is one of the well-known properties of trigonometric functions.

#### Derivative of cos(x):

The derivative of cos(x) with respect to x is -sin(x). Again, this is a well-established property of trigonometric functions.

Now that we have the derivatives of sin(x) and cos(x), we can apply the product rule:
• d(sin(x) * cos(x))/dx = (cos(x)) * cos(x) + (sin(x)) * (-sin(x))
Simplify this expression:
• cos^2(x) - sin^2(x)
Using the trigonometric identity cos^2(x) - sin^2(x) = cos2(x), we can rewrite cos^2(x) - sin^2(x)
• cos2(x)
So, the derivative of sin(x) * cos(x) is cos2(x).

### Conclusion

In this article, we've explored the process of finding the derivative of sin(x) * cos(x) using the product rule in calculus. The correct derivative is cos2(x), which is a valuable result in calculus and has applications in various fields of science and engineering. Understanding how to find derivatives of trigonometric functions and applying fundamental rules like the product rule is essential for solving more complex problems in calculus and related areas of mathematics.