Trigonometric Identities

Section 5.1 Trigonometric Identities

Goal:

T: Be able to prove trigonometric identities.
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In the Right Triangle Trigonometry section, we defined a trigonometric identity as an trigonometric equation that is true for all values of every variable in the equation for which the equation is defined. In this section, we will see that there are many other useful trigonometric identities.

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From the unit circle, you have already seen that \(\sin ^2{x} + \cos^2 {x} = 1\text{.}\) This is one of the most fundamental and most used trigonometric identities, but by the definition above, there are many more identities. Some of these can be seen easily using functions.

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Problem 5.1.1.

Consider the Pythagorean Identity \(\sin^2{x} + \cos^2{x} = 1\text{.}\)

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Divide all terms in the identity \(\sin^2{x} + \cos^2{x} = 1\) by \(\sin^2x\text{.}\) Write each quotient as a single term.
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Divide all terms in the identity \(\sin^2{x} + \cos^2{x} = 1\) by \(\cos^2x\text{.}\) Write each quotient as a single term.
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These are alternate forms of the Pythagorean Identity. Make sure you add them to your reference sheet.
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Problem 5.1.2.

One way to verify that identities are true is to use Desmos.

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What does it mean for two functions to ’line up’? How does this show that the functions are equivalent?
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Use Desmos to verify the following identities:
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1. \(\displaystyle \sin{(-x)}=-\sin{(x)}\)
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2. \(\displaystyle \cos{(-x)}=\cos{(x)}\)
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3. \(\displaystyle \tan{(-x)}=-\tan{(x)}\)
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4. \(\displaystyle \csc{(-x)}=-\csc{(x)}\)
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5. \(\displaystyle \sec{(-x)}=\sec{(x)}\)
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6. \(\displaystyle \cot{(-x)}=-\cot{(x)}\)
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7. In general, functions are called even if \(f(-x)=f(x)\text{,}\) and odd if \(f(-x)=-f(x)\text{.}\)
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  Write ’even’ or ’odd’ next to each function above.
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There are many other useful trigonometric identities. See http://www.pleacher.com/mp/mlessons/trig/ident2.html for a more complete list.In addition to using function graphs to show that trigonometric identities are true, we can also use previously proved identities and algebra.

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Example 5.1.3.

Prove that \(\tan{x} \sin{x} + \cos{x} = \sec{x}\text{.}\) Choose one side of the equation and transform it into the other side using algebra and previously used identities. NOTE: You may be tempted to perform operations (addition, subtraction, multiplication or division) to both sides of the equation. If you do this, you are using properties of equality. You are using the fact that the equation is true before you have proved that it is!

\begin{equation*} \begin{aligned} \tan x \sin x + \cos x &= \left(\frac{\sin x}{\cos x}\right)\sin x + \cos x \\ &= \frac{\sin^2 x}{\cos x} + \cos x \\ &= \frac{\sin^2 x}{\cos x} + \frac{\cos^2 x}{\cos x} \\ &= \frac{\sin^2 x + \cos^2 x}{\cos x} \\ &= \frac{1}{\cos x} \\ &= \sec x \end{aligned} \end{equation*}

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Problem 5.1.4.

Prove the following trigonometric equations using the method in Example 5.1.3.

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\(\displaystyle \csc{x} \cos{x} = \cot{x}\)
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\(\displaystyle \frac{\tan{x}} {\csc{x}} = \frac{\sin^2{x}} {\cos{x}}\)
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\(\displaystyle \sec{x} - \tan{x} = \frac{1- \sin{x}} {\cos{x}}\)
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\(\displaystyle \frac{1}{\tan{x}} + {\tan{x}} = \frac{\csc{x}} {\cos{x}}\)
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\(\displaystyle \frac{\csc x}{1-\csc x}=\frac{1}{\sin x-1}\)
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\(\displaystyle \frac{\sec x}{1+\tan^2x}=\cos x\)
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\(\displaystyle (1-\cos x)(1+\cos(-x))=\sin^2 x\)
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