In this explainer, we will learn how to simplify a trigonometric expression.

These expressions are often simplified upon the application of one or more trigonometric identity, which relate the different trigonometric and reciprocal trigonometric functions and their arguments. Their motivation is mathematical, but they also have applications in real-world problems.

Trigonometric identities have several real-world applications in various fields, such as physics, engineering, architecture, robotics, music theory, and navigation, to name a few. In physics, they can be used in projectile motion, modeling the mechanics of electromagnetic waves, analyzing alternating and direct currents, and finding the trajectory of a mass around a massive body under the force of gravity.

Let us begin by recalling the trigonometric functions, whose Pythagorean identities we will examine in this explainer. Consider the following right triangle.

The trigonometric functions can be expressed in terms of the ratio of the sides of the triangle as

These functions satisfy the following trigonometric identity:

We note that these trigonometric ratios are defined for acute angles , and the trigonometric functions for all values of are defined on the unit circle.

Suppose that a point moves along the unit circle in the counterclockwise direction. At a particular position on the unit circle with angle , the sine function is defined as and the cosine function as , as shown in the diagram above. In other words, the trigonometric functions are defined using the coordinates of the point of intersection of the unit circle with the terminal side of in the standard position.

The reciprocal trigonometric equations are defined in terms of the standard trigonometric equations as follows.

### Definition: Reciprocal Trigonometric Functions

The cosecant, secant, and cotangent functions are defined as

The trigonometric functions are periodic, which means if we add an integer multiple of , in radians, or to the angle , the value of the function stays the same:

We can see these directly from the unit circle definition of the trigonometric functions. In fact, the tangent function is periodic by , in radians, or since we have

Similarly, for the reciprocal trigonometric functions, we have

Similarly to the tangent function, the cotangent function is periodic by , in radians, or since we have

The trigonometric identities we will cover in this explainer hold for any angle in the domain of the functions, in degrees or radians. In particular, we can convert an angle between degrees and radians using the following rule: if we have angle , we can convert it to radians via

When dealing with trigonometric expressions, it is useful to rewrite the reciprocal trigonometric identities in terms of sine and cosine in order to simplify.

Letβs consider an example where we have to use reciprocal trigonometric functions to determine the value of a trigonometric expression.

### Example 1: Using Reciprocal Identities to Evaluate Trigonometric Expressions

Find the value of .

### Answer

In this example, we want to find the value a particular expression involving trigonometric and reciprocal trigonometric functions.

One way of evaluating a trigonometric expression is to write it in terms of the sine and cosine functions using the following definition for the cosecant function, which appears in the given expression:

Therefore, the expression can be simplified to give

Now, letβs consider an example where we simplify a particular trigonometric expression.

### Example 2: Simplifying Trigonometric Expressions Using Trigonometric Identities

Simplify .

### Answer

In this example, we want to simplify a particular expression involving trigonometric and reciprocal trigonometric functions.

One way of simplifying a trigonometric expression is to write it in terms of the sine and cosine functions using the following definition for the cosecant function, which appears in the given expression:

The given trigonometric expression becomes

In the next example, we will simplify a trigonometric expression by writing it out in terms of the sine and cosine functions.

### Example 3: Simplifying Trigonometric Expressions Using Trigonometric Identities

Simplify .

### Answer

In this example, we want to simplify a particular expression involving trigonometric and reciprocal trigonometric functions.

One way of simplifying a trigonometric expression is to write it in terms of the sine and cosine functions using the following definitions for the tangent and secant functions, which appear in the given expression:

The given trigonometric expression becomes

The next example involves a product of trigonometric and reciprocal trigonometric functions, which we can simply using the definition of the reciprocal functions and then rewriting the final answer in terms of another reciprocal function.

### Example 4: Simplifying Trigonometric Expressions Using Reciprocal Identities

Simplify .

### Answer

In this example, we want to simplify a particular expression involving trigonometric and reciprocal trigonometric functions.

One way of simplifying a trigonometric expression is to write it in terms of the sine and cosine functions using the following definitions for the cosecant and secant functions, which appear in the given expression:

Therefore, the expression can be simplified as

Now, using the definition of the cotangent function,

The given expression can be expressed in terms of the cotangent function as

The trigonometric and reciprocal trigonometric functions are even and odd functions as they satisfy the properties , for even functions, and , for odd functions. In particular, the sine function is odd, while the cosine function is even, since they satisfy for any value of in either degrees or radians. From this, we can also determine the parity of the other trigonometric functions that are defined in terms of these. In particular, for the tangent function, we have

Thus, the tangent function is odd, and we can deduce the parity of the other trigonometric functions in a similar manner. We can summarize these as follows.

### Even and Odd Identities for Trigonometric Functions

The cosine and secant functions are even, which means for any value of in their respective domains, they satisfy the identities

And the sine, tangent, cosecant, and cotangent functions are odd, which means they satisfy the following identities for any value of in their respective domains:

Now, letβs consider an example where we have to apply the parity of a trigonometric function in order to simply a particular trigonometric expression.

### Example 5: Simplifying Trigonometric Expressions Involving Odd and Even Identities

Simplify .

### Answer

In this example, we want to simplify a particular expression involving trigonometric and reciprocal trigonometric functions using an even/odd identity.

One way of simplifying a trigonometric expression is to write it in terms of the sine and cosine functions using the following definition for the cosecant function, which appears in the given expression:

The tangent function is odd, hence the identity

We can rewrite the tangent function using its definition in terms of the sine and cosine functions:

Therefore, the expression can be simplified as

Finally, we can rewrite this expression in terms of the secant function defined as

Thus, the expression becomes

The sine function is equivalent to the cosine function by a translation to the left, which can be visualized by comparing the plot of both functions.

In particular, we have the following shift identities for the angles and :

We can also illustrate these on the unit circle as shown.

Similarly, by replacing with , we obtain the following cofunction identities for the complementary angles and :

We can illustrate this as shown.

The figure depicts a right triangle with angle in standard position, which intersects the unit circle at and has an acute-angle measure .

We can combine these identities and use them to determine identities for the other trigonometric functions that are defined in terms of the sine and cosine functions.

### Definition: Trigonometric Correlated-Angle Identities

The trigonometric functions satisfy cofunction identities for all in their domains. In particular, we have

For example, for the tangent function, we have

All of these identities also hold in radians, in particular, by replacing with in radians.

Now, letβs consider an example where we use this identity along with the parity of a trigonometric function in order to simplify an expression.

### Example 6: Simplifying Trigonometric Expressions Using Correlated and Even Identities

Simplify .

### Answer

In this example, we want to simplify a particular expression involving reciprocal trigonometric functions.

We will also make use of the correlated-angle identity and the even identity

One way of simplifying a trigonometric expression is to write it in terms of the sine and cosine functions using the following definition for the secant function:

Using these, the expression becomes

Now, suppose we want to determine . We can find this by repeatedly using the identities above. If we let , then

Now, substituting back , we obtain

Similarly, we find

By repeatedly applying these identities or using the unit circle, we also have the identities for the angles and :

For and , we have the following.

And for and , we have the following.

We also have identities for the other trigonometric functions, which follow from those for the sine and cosine functions, from their definitions:

The next example involves using the definitions of the reciprocal trigonometric functions along with the cofunction identities in radians in order to simplify an expression.

### Example 7: Using Periodic and Cofunction Identities to Simplify a Trigonometric Expression

Simplify .

### Answer

In this example, we want to simplify a particular expression involving reciprocal trigonometric functions.

We will also make use of the cofunction and correlated identities:

One way of simplifying a trigonometric expression is to write it in terms of the sine and cosine functions using the following definitions for the cosecant and cotangent functions, which appear in the numerator and denominator:

The numerator of the expression can be simplified as

And the denominator as

Therefore, the expression can be simplified as

Finally, we can rewrite this expression in terms of the secant function defined as

Thus, we obtain

Similarly, for the angles and , we have

This can be visualized as follows.

By using the periodicity of the trigonometric functions and the unit circle, we have

All of the identities also hold in radians, by replacing with in radians. They can also be visualized using the unit circle as shown.

All of the correlated-angle identities can be visualized using the following.

Now, let us consider a few examples where we have to apply the cofunction identities in order to simplify a trigonometric expression. In the next example, we will use this identity repeatedly on the cosine and sine functions, in degrees.

### Example 8: Simplifying Trigonometric Expressions Using Cofunction Identities

Simplify .

### Answer

In this example, we want to simplify a particular expression involving trigonometric functions.

In order to simplify the given expression, we make use of the correlated-angle identity

Therefore, we have

In the last example, we want to apply the cofunction identities to the tangent and cotangent functions repeatedly, in degrees, in order to simplify a trigonometric expression.

### Example 9: Using Trigonometric Identities to Simplify a Trigonometric Expression

Simplify .

### Answer

We will also make use of the correlated-angle identity

Since we have, by definition of the cotangent function, the correlated identity can be written in terms of the tangent function as

Therefore, the expression can be simplified as

Let us finish by recapping a few important key points from this explainer.

### Key Points

- We can express the tangent and reciprocal trigonometric functions in terms of sine and cosine as and we can use these to simplify trigonometric expressions.
- These trigonometric functions are all either even or odd. In particular, for the sine and cosine functions, we have and similarly for the other trigonometric functions that follow from the definitions. We can use the parity of the trigonometric functions to help us simplify trigonometric expressions.
- The unit circle allows us to determine the correlated-angle identities for sine and cosine.

For instance, the cofunction identities (in radians) are, The corresponding identities for the tangent and reciprocal trigonometric functions are found using their definitions in terms of the sine and cosine functions. - We often need to apply more than one identity, or type of identity, to simplify a trigonometric expression.