# 9.2 Sum and difference identities  (Page 4/6)

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## Using sum and difference formulas for cofunctions

Now that we can find the sine, cosine, and tangent functions for the sums and differences of angles, we can use them to do the same for their cofunctions. You may recall from Right Triangle Trigonometry that, if the sum of two positive angles is $\text{\hspace{0.17em}}\frac{\pi }{2},$ those two angles are complements, and the sum of the two acute angles in a right triangle is $\text{\hspace{0.17em}}\frac{\pi }{2},$ so they are also complements. In [link] , notice that if one of the acute angles is labeled as $\text{\hspace{0.17em}}\theta ,$ then the other acute angle must be labeled $\text{\hspace{0.17em}}\left(\frac{\pi }{2}-\theta \right).$

Notice also that $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta =\mathrm{cos}\left(\frac{\pi }{2}-\theta \right),$ which is opposite over hypotenuse. Thus, when two angles are complimentary, we can say that the sine of $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ equals the cofunction of the complement of $\text{\hspace{0.17em}}\theta .\text{\hspace{0.17em}}$ Similarly, tangent and cotangent are cofunctions, and secant and cosecant are cofunctions.

From these relationships, the cofunction identities are formed. Recall that you first encountered these identities in The Unit Circle: Sine and Cosine Functions .

## Cofunction identities

The cofunction identities are summarized in [link] .

 $\mathrm{sin}\text{\hspace{0.17em}}\theta =\mathrm{cos}\left(\frac{\pi }{2}-\theta \right)$ $\mathrm{cos}\text{\hspace{0.17em}}\theta =\mathrm{sin}\left(\frac{\pi }{2}-\theta \right)$ $\mathrm{tan}\text{\hspace{0.17em}}\theta =\mathrm{cot}\left(\frac{\pi }{2}-\theta \right)$ $\mathrm{cot}\text{\hspace{0.17em}}\theta =\mathrm{tan}\left(\frac{\pi }{2}-\theta \right)$ $\mathrm{sec}\text{\hspace{0.17em}}\theta =\mathrm{csc}\left(\frac{\pi }{2}-\theta \right)$ $\mathrm{csc}\text{\hspace{0.17em}}\theta =\mathrm{sec}\left(\frac{\pi }{2}-\theta \right)$

Notice that the formulas in the table may also justified algebraically using the sum and difference formulas. For example, using

$\mathrm{cos}\left(\alpha -\beta \right)=\mathrm{cos}\text{\hspace{0.17em}}\alpha \mathrm{cos}\text{\hspace{0.17em}}\beta +\mathrm{sin}\text{\hspace{0.17em}}\alpha \mathrm{sin}\text{\hspace{0.17em}}\beta ,$

we can write

$\begin{array}{ccc}\hfill \mathrm{cos}\left(\frac{\pi }{2}-\theta \right)& =& \mathrm{cos}\text{\hspace{0.17em}}\frac{\pi }{2}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta +\mathrm{sin}\text{\hspace{0.17em}}\frac{\pi }{2}\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta \hfill \\ & =& \left(0\right)\mathrm{cos}\text{\hspace{0.17em}}\theta +\left(1\right)\mathrm{sin}\text{\hspace{0.17em}}\theta \hfill \\ & =& \mathrm{sin}\text{\hspace{0.17em}}\theta \hfill \end{array}$

## Finding a cofunction with the same value as the given expression

Write $\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\frac{\pi }{9}\text{\hspace{0.17em}}$ in terms of its cofunction.

The cofunction of $\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\theta =\mathrm{cot}\left(\frac{\pi }{2}-\theta \right).\text{\hspace{0.17em}}$ Thus,

$\begin{array}{ccc}\hfill \mathrm{tan}\left(\frac{\pi }{9}\right)& =& \mathrm{cot}\left(\frac{\pi }{2}-\frac{\pi }{9}\right)\hfill \\ & =& \mathrm{cot}\left(\frac{9\pi }{18}-\frac{2\pi }{18}\right)\hfill \\ & =& \mathrm{cot}\left(\frac{7\pi }{18}\right)\hfill \end{array}$

Write $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\frac{\pi }{7}\text{\hspace{0.17em}}$ in terms of its cofunction.

$\mathrm{cos}\left(\frac{5\pi }{14}\right)$

## Using the sum and difference formulas to verify identities

Verifying an identity means demonstrating that the equation holds for all values of the variable. It helps to be very familiar with the identities or to have a list of them accessible while working the problems. Reviewing the general rules presented earlier may help simplify the process of verifying an identity.

Given an identity, verify using sum and difference formulas.

1. Begin with the expression on the side of the equal sign that appears most complex. Rewrite that expression until it matches the other side of the equal sign. Occasionally, we might have to alter both sides, but working on only one side is the most efficient.
2. Look for opportunities to use the sum and difference formulas.
3. Rewrite sums or differences of quotients as single quotients.
4. If the process becomes cumbersome, rewrite the expression in terms of sines and cosines.

## Verifying an identity involving sine

Verify the identity $\text{\hspace{0.17em}}\mathrm{sin}\left(\alpha +\beta \right)+\mathrm{sin}\left(\alpha -\beta \right)=2\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta .$

We see that the left side of the equation includes the sines of the sum and the difference of angles.

$\begin{array}{ccc}\hfill \mathrm{sin}\left(\alpha +\beta \right)& =& \mathrm{sin}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta +\mathrm{cos}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\beta \hfill \\ \hfill \mathrm{sin}\left(\alpha -\beta \right)& =& \mathrm{sin}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta -\mathrm{cos}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\beta \hfill \end{array}$

We can rewrite each using the sum and difference formulas.

$\begin{array}{ccc}\hfill \mathrm{sin}\left(\alpha +\beta \right)+\mathrm{sin}\left(\alpha -\beta \right)& =& \mathrm{sin}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta +\mathrm{cos}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\beta +\mathrm{sin}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta -\mathrm{cos}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\beta \hfill \\ & =& 2\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta \hfill \end{array}$

We see that the identity is verified.

## Verifying an identity involving tangent

Verify the following identity.

$\frac{\mathrm{sin}\left(\alpha -\beta \right)}{\mathrm{cos}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta }=\mathrm{tan}\text{\hspace{0.17em}}\alpha -\mathrm{tan}\text{\hspace{0.17em}}\beta$

We can begin by rewriting the numerator on the left side of the equation.

We see that the identity is verified. In many cases, verifying tangent identities can successfully be accomplished by writing the tangent in terms of sine and cosine.

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