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4.1: Trigonometric Identities

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Focus Questions

The following questions are meant to guide our study of the material in this section. After studying this section, we should understand the concepts mo- tivated by these questions and be able to write precise, coherent answers to these questions.

• What is an identity?
• How do we verify an identity?

Consider the trigonometric equation $$\sin(2x) = \cos(x)$$. Based on our current knowledge, an equation like this can be difficult to solve exactly because the periods of the functions involved are different. What will allow us to solve this equation relatively easily is a trigonometric identity, and we will explicitly solve this equation in a subsequent section. This section is an introduction to trigonometric identities.

As we discussed in Section 2.6, a mathematical equation like $$x^{2} = 1$$ is a relation between two expressions that may be true for some values of the variable. To solve an equation means to find all of the values for the variables that make the two expressions equal to each other. An identity, is an equation that is true for all allowable values of the variable. For example, from previous algebra courses, we have seen that

$x^{2} - 1 = (x + 1)(x - 1)$

for all real numbers $$x$$. This is an algebraic identity since it is true for all real number values of $$x$$. An example of a trigonometric identity is $$\cos^{2} + \sin^{2} = 1$$ since this is true for all real number values of $$x$$.

So while we solve equations to determine when the equality is valid, there is no reason to solve an identity since the equality in an identity is always valid. Every identity is an equation, but not every equation is an identity. To know that an equation is an identity it is necessary to provide a convincing argument that the two expressions in the equation are always equal to each other. Such a convincing argument is called a proof and we use proofs to verify trigonometric identities.

Definition: Identity

An identity is an equation that is true for all allowable values of the variables involved.

Beginning Activity

1. Use a graphing utility to draw the graph of $$y = \cos(x - \dfrac{\pi}{2})$$ and $$y = \sin(x + \dfrac{\pi}{2})$$ over the interval $$[-2\pi, 2\pi]$$ on the same set of axes. Are the two expressions $$\cos(x - \dfrac{\pi}{2})$$ and $$\sin(x + \dfrac{\pi}{2})$$ the same – that is, do they have the same value for every input $$x$$? If so, explain how the graphs indicate that the expressions are the same. If not, find at least one value of $$x$$ at which $$\cos(x - \dfrac{\pi}{2})$$ and $$y = \sin(x + \dfrac{\pi}{2})$$ have different values.
2. Use a graphing utility to draw the graph of $$y = \cos(x - \dfrac{\pi}{2})$$ and $$y = \sin(x)$$ over the interval $$[-2\pi , 2\pi]$$ on the same set of axes. Are the two expressions $$\cos(x - \dfrac{\pi}{2})$$ and $$\sin(x)$$ the same – that is, do they have the same value for every input $$x$$? If so, explain how the graphs indicate that the expressions are the same. If not, find at least one value of $$x$$ at which $$\cos(x - \dfrac{\pi}{2})$$ and $$\sin(x)$$ have different values.

Some Known Trigonometric Identities

We have already established some important trigonometric identities. We can use the following identities to help establish new identities.

The Pythagorean Identity

This identity is fundamental to the development of trigonometry. See Section 1.2.

For all real numbers $$t$$,

$\cos^{2} + \sin^{2} = 1.$

Identities from Definitions

The definitions of the tangent, cotangent, secant, and cosecant functions were introduced in Section 1.6. The following are valid for all values of $$t$$ for which the right side of each equation is defined.

$\tan(t) = \dfrac{\sin(t)}{\cos(t)}$

$\cot(t) = \dfrac{\cos(t)}{\sin(t)}$

$\sec(t) = \dfrac{1}{\cos(t)}$

$\csc(t) = \dfrac{1}{\sin(t)}$

Negative Identities

The negative were introduced in Chapter 2 when the symmetry of the graphs were discussed. (See page 82 and Exercise (2) on page 139.)

$\cos(-t) = \cos(t)$

$\sin(-t) = -\sin(t)$

$\tan(-t) = -\tan(t)$

The negative identities for cosine and sine are valid for all real numbers $$t$$, and the negative identity for tangent is valid for all real numbers $$t$$ for which $$\tan(t)$$ is defined.

Verifying Identities

Given two expressions, say $$\tan^{2}(x) + 1$$ and $$\sec^{2}(x)$$, we would like to know if they are equal (that is, have the same values for every allowable input) or not. We can draw the graphs of $$y = \tan^{2}(x) + 1$$ and $$y = \sec^{2}(x)$$ and see if the graphs look the same or different. Even if the graphs look the same, as they do with $$y = \tan^{2}(x) + 1$$ and $$y = \sec^{2}(x)$$, that is only an indication that the two expressions are equal for every allowable input. In order to verify that the expressions are in fact always equal, we need to provide a convincing argument that works for all possible input. To do so we use facts that we know (existing identities) to show that two trigonometric expressions are always equal. As an example, we will verify that the equation $\tan^{2}(x) + 1 = \sec^{2}(x)$ is an identity.

A proper format for this kind of argument is to choose one side of the equation and apply existing identities that we already know to transform the chosen side into the remaining side. It usually makes life easier to begin with the more complicated looking side (if there is one). In our example of equation (1) we might begin with the expression $$\tan^{2}(x) + 1$$.

Example $$\PageIndex{1}$$: Verifying a Trigonometric Identity

To verify that equation (1) is an identity, we work with the expression $$\tan^{2}(x) + 1$$. It can often be a good idea to write all of the trigonometric functions in terms of the cosine and sine to start. In this case, we know that $$\tan(t) = \dfrac{\sin(t)}{\cos(t)}$$, so we could begin by making this substitution to obtain the identity $\tan^{2}(x) + 1 = (\dfrac{\sin(x)}{\cos(x)})^{2} + 1$

Note that this is an identity and so is valid for all allowable values of the variable. Next we can apply the square to both the numerator and denominator of the right hand side of our identity (2).

$(\dfrac{\sin(x)}{\cos(x)})^{2} + 1 = \dfrac{\sin^{2}(x)}{\cos^{2}(x)} + 1$

Next we can perform some algebra to combine the two fractions on the right hand side of the identity (3) and obtain the new identity

$\dfrac{\sin^{2}(x)}{\cos^{2}(x)} + 1 = \dfrac{\sin^{2}(x) + \cos^{2}(x)}{\cos^{2}(x)}$

Now we can recognize the Pythagorean identity $$\cos^{2}(x) + \sin^{2}(x) = 1$$, which makes the right side of identity (4)
$\dfrac{\sin^{2}(x) + \cos^{2}(x)}{\cos^{2}(x)} = \dfrac{1}{\cos^{2}(x)}$

Recall that our goal is to verify identity (1), so we need to transform the expression into $$\sec^{2}(x)$$. Recall that $$\sec(x) = \dfrac{1}{\cos(x)}$$, and so the right side of identity (5) leads to the new identity which verifies the identity.

$\dfrac{1}{\cos^{2}(x)} = \sec^{2}(x)$

An argument like the one we just gave that shows that an equation is an identity is called a proof. We usually leave out most of the explanatory steps (the steps should be evident from the equations) and write a proof in one long string of identities as

$\tan^{2}(x) + 1 = (\dfrac{\sin(x)}{\cos(x)})^{2} + 1 = \dfrac{\sin^{2}(x)}{\cos^{2}(x)} + 1= \dfrac{\sin^{2} + \cos^{2}(x)}{\cos^{2}(x)} = \dfrac{1}{\cos^{2}(x)} = \sec^{2}(x).$

To prove an identity is to show that the expressions on each side of the equation are the same for every allowable input. We illustrated this process with the equation $$\tan^{2}(x) + 1 = \sec^{2}(x)$$. To show that an equation isn’t an identity it is enough to demonstrate that the two sides of the equation have different values at one input.

Example $$\PageIndex{2}$$: (Showing that an Equation is not an Identity)

Consider the equation with the equation $$\cos(x - \dfrac{\pi}{2}) = \sin(x + \dfrac{\pi}{2})$$ that we encountered in our Beginning Activity. Although you can check that $$\cos(x - \dfrac{\pi}{2})$$ and $$\sin(x + \dfrac{\pi}{2})$$ are equal at some values, $$\dfrac{\pi}{4}$$ for example, they are not equal at all values–$$\cos(0 - \dfrac{\pi}{2}) = 0$$ but $$\sin(0 + \dfrac{\pi}{2}) = 1$$. Since an identity must provide an equality for all allowable values of the variable, if the two expressions differ at one input, then the equation is not an identity. So the equation $$\cos(x - \dfrac{\pi}{2}) = \sin(x + \dfrac{\pi}{2})$$ is not an identity.

Example 4.2 illustrates an important point. to show that an equation is not an identity, it is enough to find one input at which the two sides of the equation are not equal. We summarize our work with identities as follows.

• To prove that an equation is an identity, we need to apply known identities to show that one side of the equation can be transformed into the other.
• To prove that an equation is not an identity, we need to find one input at which the two sides of the equation have different values.

Important Note: When proving an identity it might be tempting to start working with the equation itself and manipulate both sides until you arrive at something you know to be true. DO NOT DO THIS! By working with both sides of the equation, we are making the assumption that the equation is an identity – but this assumes the very thing we need to show. So the proper format for a proof of a trigonometric identity is to choose one side of the equation and apply existing identities that we already know to transform the chosen side into the remaining side. It usually makes life easier to begin with the more complicated looking side (if there is one).

Example $$\PageIndex{3}$$: Verifying an Identity

Consider the equation $2\cos^{2}(x) - 1 = \cos^{2}(x) - \sin^{2}(x).$
Graphs of both sides appear to indicate that this equation is an identity. To prove the identity we start with the left hand side:

$2\cos^{2}(x) - 1 = \cos^{2}(x) + \cos^{2}(x) - 1 = \cos^{2}(x) + (1 - \sin^{2}(x)) - 1 = \cos^{2}(x) - \sin^{2}(x).$

Notice that in our proof we rewrote the Pythagorean identity $$\cos^{2}(x) + \sin^{2}(x) = 1$$ as $$\cos^{2}(x) = 1 - \sin^{2}(x)$$. Any proper rearrangement of an identity is also an identity, so we can manipulate known identities to use in our proofs as well.

To reiterate, the proper format for a proof of a trigonometric identity is to choose one side of the equation and apply existing identities that we already know to transform the chosen side into the remaining side. There are no hard and fast methods for proving identities – it is a bit of an art. You must practice to become good at it.

Exercise $$\PageIndex{1}$$

For each of the following use a graphing utility to graph both sides of the equation. If the graphs indicate that the equation is not an identity, find one value of $$x$$ at which the two sides of the equation have different values. If the graphs indicate that the equation is an identity, verify the identity.

1. $\dfrac{\sec^{2}(x) - 1}{\sec^{2}(x)} = \sin^{2}(x)$
2. $\cos(x)\sin(x) = 2\sin(x)$

1. The graphs of both sides of the equation indicate that this is an indentity.

2. The graphs of both sides of the equation indicate that this is not an indentity. For example, if we let $$x = \dfrac{\pi}{2}$$,then

$\cos(\dfrac{\pi}{2})\sin(\dfrac{\pi}{2}) = 0\cdot 1 = 0$ and $2\sin(\dfrac{\pi}{2}) = 2\cdot 1 = 2$

Summary

In this section, we studied the following important concepts and ideas:

An identity is an equation that is true for all allowable values of the variables involved.

• To prove that an equation is an identity, we need to apply known identities to show that one side of the equation can be transformed into the other.
• To prove that an equation is not an identity, we need to find one input at which the two sides of the equation have different values.