Verify that each equation is an identity. - Numerade 6.3: Verifying Trigonometric Identities is shared under a CC BY license and was authored, remixed, and/or curated by LibreTexts. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. This problem illustrates that there are multiple ways we can verify an identity. = \tan^2 \theta\cdot \cos^2 \theta &&\text{Use QuotientIdentity: } \tan\theta = \dfrac{\sin \theta}{\cos\theta}\\[2pt] The approach to verifying an identity depends on the nature of the identity. There are no hard and fast methods for proving identities it is a bit of an art. = \dfrac{\sin \theta}{\cos \theta} \cdot \dfrac{\cos \theta }{1} & &\text{Cancel} \\[2pt] step1: reciprocal-------------because cotx=1 View the full answer Transcribed image text: Verify that the equation is an identity. Step 1 of 4. Explanation: Left Side = (SecA - TanA) (SecA + TanA) = sec2A +secAtanA tanAsecA tan2A Notice that secAtanA tanAsecA = 0 So Left Side = sec2A tan2A Now apply the Pythagorean Identity tan2A+ 1 = sec2A by replacing the sec2A by tan2A+ 1 Left Side = tan2A+ 1 tan2A Left Side = 1 Left Side = Right Side [Q.E.D.] verify that each equation is an identity and show all work This problem has been solved! Q: Verify that each equation is an identity 2 / 1 + cos x - tan^2 x/2 = 1. It usually makes life easier to begin with the more complicated looking side (if there is one). This problem illustrates that there are multiple ways we can verify an identity. What is an identity, and how do I prove it? | Purplemath \end{array} \), Example \(\PageIndex{9}\): Verifya Trigonometric Identity - Factor, \(\dfrac{{\sin}^2(\theta){\cos}^2(\theta)}{\sin(\theta)\cos(\theta)}=\cos \theta\sin \theta\). 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. = 1 - \sin x + \sin x -\sin^2 x & &\text{Combine like terms} \\[2pt] The Pythagorean Identities are based on the properties of a right triangle. 6. Lets start with the left side and simplify: \( \begin{array} {l|ll } Describe how to manipulate the equations to get from [latex]{\sin }^{2}t+{\cos }^{2}t=1[/latex] to the other forms. 1/1-sin x + 1/1+sin x = 2/cos^2x Simplifying one side of the equation to equal the other side is another method for verifying an identity. We will discuss the role of identities more after an example. With a combination of tangent and sine, we might try rewriting tangent, \(\tan (x)=3\sin (x)\) We can interpret the tangent of a negative angle as [latex]\tan \left(-\theta \right)=\frac{\sin \left(-\theta \right)}{\cos \left(-\theta \right)}=\frac{-\sin \theta }{\cos \theta }=-\tan \theta[/latex]. Verify the equation is an identity | Wyzant Ask An Expert Verify that each equation is an identity. The Reciprocal Identities define reciprocals of the trigonometric functions. (Hint: Multiply the numerator and denominator on the left side by [latex]1-\sin\theta[/latex]). The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Solved Verify that the equation is an identity. (Hint: cos2x - Chegg Verify that each equation is an identity. $$ \sin ( A + B | Quizlet This does not appear to factor nicely so we use the quadratic formula, remembering that we are solving for cos( \(\theta\)). Graphing both sides of an identity will verify it. Graphs of both sides appear to indicate that this equation is an identity. We can start with the Pythagorean identity. \[\dfrac{\sin (x)}{\cos (x)} =3\sin (x)\nonumber\], Multiplying both sides by cosine &= 1+{\cot}^2 \theta-{\cot}^2 \theta\\ This is the difference of squares. Create an identity for the expression [latex]2\tan \theta \sec \theta[/latex] by rewriting strictly in terms of sine. To verify the trigonometric identities, we usually start with the more complicated side of the equation and essentially rewrite the expression until it has been transformed into the same expression as the other side of the equation. \dfrac{{\sec}^2 \theta1}{\sec^2 \theta} &={\sin}^2 \theta&\text{Use the Pythagorean Identity: } \tan ^2 (\theta )+1=\sec ^{2} (\theta )\\[2pt] sin ( 4 ) + sin ( 8 ) sin ( 4 ) sin ( 8 ) = tan ( 6 ) tan ( 2 ) 01:09 Verify that each equation is an identity. \( \cos (\theta )=\pm \sqrt{\dfrac{1}{2} } =\pm \dfrac{\sqrt{2} }{2} \) Solution Verified Step 1 1 of 3 Since we need to prove the identity 1+tanxtan2x=sec2x\color{#c34632}1+\tan x\tan2x=\sec2x1+tanxtan2x=sec2x, we can simplify the left side as follows: \[\theta =\cos ^{-1} \left(0.425\right)=1.131\nonumber\] By symmetry, a second solution can be found Verify the equation is an identity The question states: To verify the identity, start with the more complicated side and transform it to look like the other side. If both expressions give the same graph, then they must be identities. \end{array} \). We have seen that algebra is very important in verifying trigonometric identities, but it is just as critical in simplifying trigonometric expressions before solving. 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. Answered: Verify that each equation is an | bartleby Graph of [latex]y=\sin \theta[/latex], Recall that an even function is one in which. Since the left side seems a bit more complicated, we will start there and simplify the expression until we obtain the right side. To verify an identity, we show that ______ side of the identity can be simplified so that it is identical to the other side. We have four solutions on \(0 \le x<2\pi\): \[x = 0, 1.231, \quad\pi , 5.052\nonumber\], Example \(\PageIndex{17}\): Use Reciprocal Identities. (Tan x / 1+ cos x) + (sin x / 1-cos x) = cot x + sec x csc x (1- sin / 1+sin) = sec^2 - 2sectan + tan^2 Expert Solution Trending now This is a popular solution! The cosine function is an even function because [latex]\cos \left(-\theta \right)=\cos \theta[/latex]. So the equation \(\cos(x - \dfrac{\pi}{2}) = \sin(x + \dfrac{\pi}{2})\) is not an identity. An argument like the one we just gave that shows that an equation is an identity is called a proof. We can use the right side as a guide for what might be good steps to make. In addition to the Pythagorean Identity, it is often necessary to rewrite the tangent, secant, cosecant, and cotangent as part of solving an equation. Prove the identity solver - SOFTMATH To avoid this problem, we can rearrange the equation so that one side is zero (You technically can divide by sin(x), as long as you separately consider the case where sin(x) = 0. [latex]\frac{\tan x+\cot x}{\csc x};\cos x[/latex], 17. 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. [latex]\frac{1-{\cos }^{2}x}{{\tan }^{2}x}+2{\sin }^{2}x[/latex]. \[x=2\pi -1.231=5.052 \nonumber\]. \( \begin{array} {lll } The sine function is an odd function because [latex]\sin \left(-\theta \right)=-\sin \theta[/latex]. = ({\sin}^2 x)\left (\dfrac{1}{{\sin}^2 x}\right ) &&\text{Cancel} \\ As the left side is more complicated, lets begin there. = \cos^2 x \;\;\color{Cerulean}{} & &\text{Establish the identity} \\ After examining the reciprocal identity for [latex]\sec t[/latex], explain why the function is undefined at certain points. We see only one graph because both expressions generate the same image. See Section 1.2. Solved: For the following exercises, verify that each equation is Simplify the expression by rewriting and using identities: [latex]{\csc }^{2}\theta -{\cot }^{2}\theta [/latex]. Using algebra makes finding a solution straightforward and familiar. It is usually better to start with the more complex side, as it is easier to simplify than to build. = \dfrac{\sin^2 \theta}{\cos^2 \theta}\cdot \cos^2 \theta &&\text{Cancel}\\[2pt] DO NOT DO THIS! Solve \(2\sin ^{2} (t)=3\cos (t)\) for all solutions with \(0\le t<2\pi\). = {\sin}^2 \theta \;\;\color{Cerulean}{}&&\text{Establish the identity} \\ For all [latex]\theta[/latex] in the domain of the sine and cosine functions, respectively, we can state the following: The other even-odd identities follow from the even and odd nature of the sine and cosine functions. Work on one side of the equation. Answer link Verify the fundamental trigonometric identities. Accessibility StatementFor more information contact us atinfo@libretexts.org. If these steps do not yield the desired result, try converting all terms to sines and cosines. Q: Find the exact value of the expression. Simplify the expression by rewriting and using identities:\({\csc}^2 \theta{\cot}^2 \theta\), \[\begin{align*} Every identity is an equation, but not every equation is an identity. = \dfrac{(\sin \theta-\cos \theta)}{-1} & &\text{ } \\ View this answer View this answer View this answer done loading. Verify \(\tan \theta \cos \theta=\sin \theta\). As the left side is more complicated, lets begin there. = 1\cdot \dfrac{\sin (\alpha )}{1} +\dfrac{\cos (\alpha )}{\sin (\alpha )} \cdot \dfrac{\sin (\alpha )}{1} & & \text{Simplifyfractions} \\[2pt]
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