1.
[latex]\left(\sqrt{a} + \sqrt{b}\right)^2 = a + b[/latex]
Chapter 5: Equations and Identities
Practice each skill in the Homework Problems listed:
For Problems 1–8, decide which of the following equations are identities. Explain your reasoning.
[latex]\left(\sqrt{a} + \sqrt{b}\right)^2 = a + b[/latex]
[latex]\sqrt{a^2 - b^2} = a - b[/latex]
[latex]\dfrac{1}{a + b} = \dfrac{1}{a} + \dfrac{1}{b}[/latex]
[latex]\dfrac{a + b}{a} = b[/latex]
[latex]\tan (\alpha + \beta) = \dfrac{\sin (\alpha + \beta)}{\cos (\alpha + \beta)}[/latex]
[latex]\dfrac{1}{\tan \theta} = \dfrac{\cos \theta}{\sin \theta}[/latex]
[latex](1 + \tan \theta)^2 = 1 + \tan^2 \theta[/latex]
[latex]\sqrt{1 - \sin^2 \phi} = 1 - \sin \phi[/latex]
For Problems 9–16, use graphs to decide which of the following equations are identities.
[latex]\sin 2t = 2 \sin t[/latex]
[latex]\cos \theta + \sin \theta = 1[/latex]
[latex]\sin (30° + \beta) = \dfrac{1}{2} + \sin \beta[/latex]
[latex]\cos (90° - C) = \sin C[/latex]
[latex]\tan (90° - \theta) = \dfrac{1}{\tan \theta}[/latex]
[latex]\tan 2\theta = \dfrac{2\tan \theta}{1 - \tan^2 \theta}[/latex]
[latex]\dfrac{\tan^2 x}{1 + \tan^2 x} = \sin^2 x[/latex]
[latex]\tan x + \dfrac{1}{\tan x} = \sin x \cos x[/latex]
For Problems 17–26, show that the equation is an identity by transforming the left side into the right side.
[latex](1 + \sin w)(1 - \sin w) = \cos^2 w[/latex]
[latex](\cos \theta - 1)(\cos \theta + 1) = -\sin^2 \theta[/latex]
[latex](\cos \theta - \sin \theta)^2 = 1 - 2 \sin \theta \cos \theta[/latex]
[latex]\sin^2 x - \cos^2 x = 1 - 2\cos^2 x[/latex]
[latex]\tan \theta \cos \theta = \sin \theta[/latex]
[latex]\dfrac{\sin \mu}{\tan \mu} = \cos \mu[/latex]
[latex]\cos^4 x - \sin^4 x = \cos^2 x - \sin^2 x[/latex]
[latex]1 - 2\cos^2 v + \cos^4 v = \sin^4 v[/latex]
[latex]\dfrac{\sin u}{1 + \cos u} = \dfrac{1 - \cos u}{\sin u}[/latex]
Multiply numerator and denominator of the left side by [latex]1 - \cos u{.}[/latex]
[latex]\dfrac{\sin v}{1 - \cos v} = \dfrac{\tan v(1 + \sin v)}{\cos v}[/latex]
Multiply numerator and denominator of the left side by [latex]1 + \sin v{.}[/latex]
For Problems 27–34, simplify, using identities as necessary.
[latex]\dfrac{1}{\cos^2 \beta}- \dfrac{\sin^2 \beta}{\cos^2 \beta}[/latex]
[latex]\dfrac{1}{\sin^2 \phi}- \dfrac{1}{\tan^2 \phi}[/latex]
[latex]\cos^2 \alpha (1 + \tan^2 \alpha)[/latex]
[latex]\cos^3 \phi + \sin^2 \phi \cos \phi[/latex]
[latex]\tan^2 A - \tan^2 A \sin^2 A[/latex]
[latex]\cos^2 B \tan^2 B + \cos^2 B[/latex]
[latex]\dfrac{1 - \cos^2 z}{\cos^2 z}[/latex]
[latex]\dfrac{\sin t}{\cos t \tan t}[/latex]
For Problems 35–40, evaluate without using a calculator.
[latex]3\cos^2 1.7° + 3\sin^2 1.7°[/latex]
[latex]4 - \cos^2 338° - \sin^2 338°[/latex]
[latex](\cos^2 20° + \sin^2 20°)^4[/latex]
[latex]\dfrac{18}{\cos^2 17° + \sin^2 17°}[/latex]
[latex]\dfrac{6}{\cos^2 53°} - 6 \tan^2 53°[/latex]
[latex]\dfrac{1}{\sin^2 102°} - \dfrac{\cos^2 102°}{\sin^2 102°}[/latex]
For Problems 41–46, one side of an identity is given. Graph the expression and make a conjecture about the other side of the identity.
[latex]2\cos^2 \theta - 1 = ?[/latex]
[latex]1 - 2\sin^2 \left(\dfrac{\theta}{2}\right) = ?[/latex]
[latex]1 - \dfrac{\sin^2 x}{1 + \cos x} = ?[/latex]
[latex]\dfrac{\sin x}{\sqrt{1 - \sin^2 x}} = ?[/latex]
[latex]2\tan t \cos^2 t = ?[/latex]
[latex]\dfrac{2 \tan t}{1 - \tan^2 t} = ?[/latex]
For Problems 47–50, use identities to rewrite each expression.
[latex]2 - \cos^2 \theta + 2 \sin \theta~~~[/latex] as an expression in [latex]\sin \theta[/latex] only
[latex]3\sin^2 B + 2\cos B - 4~~~[/latex] as an expression in [latex]\cos B[/latex] only
[latex]\cos^2 \phi - 2\sin^2 \phi~~~[/latex] as an expression in [latex]\cos \phi[/latex] only
[latex]\cos^2 \phi \sin^2 \phi~~~[/latex] as an expression in [latex]\sin \phi[/latex] only
For Problems 51–58, solve the equation for [latex]0° \le\theta\le 360°{.}[/latex] Round angles to three decimal places if necessary.
[latex]\cos \theta - \sin^2 \theta + 1 = 0[/latex]
[latex]4\sin \theta + 2\cos^2 \theta - 3 = -1[/latex]
[latex]1 - \sin \theta - 2\cos^2 \theta = 0[/latex]
[latex]3\cos^2 \theta - \sin^2 \theta = 2[/latex]
[latex]2\cos \theta \tan \theta + 1 = 0[/latex]
[latex]\cos \theta - \sin \theta = 0[/latex]
[latex]\dfrac{1}{3}\cos \theta = \sin \theta[/latex]
[latex]5\sin C = 2\cos C[/latex]
For Problems 59–62, use identities to find exact values for the other two trig ratios.
[latex]\cos A = \dfrac{12}{13}~~~[/latex] and [latex]~270° \lt\ A \lt 360°[/latex]
[latex]\sin B = \dfrac{-3}{5}~~~[/latex] and [latex]~180° \lt\ B \lt 270°[/latex]
[latex]\sin \phi = \dfrac{1}{7}~~~[/latex] and [latex]~90° \lt\ \phi \lt 180°[/latex]
[latex]\cos t = \dfrac{-2}{3}~~~[/latex] and [latex]~180° \lt\ t \lt 270°[/latex]
For Problems 63–66, use the identity below to find the sine and cosine of the angle.
[latex]{1 + \tan^2 \theta = \dfrac{1}{\cos^2 \theta}}[/latex]
[latex]\tan \theta = -\dfrac{1}{2}~~~[/latex] and [latex]~270° \lt\ \theta \lt 360°[/latex]
[latex]\tan \theta = 2~~~[/latex] and [latex]~180° \lt\ \theta \lt 270°[/latex]
[latex]\tan \theta = \dfrac{3}{4}~~~[/latex] and [latex]~180° \lt\ \theta \lt 270°[/latex]
[latex]\tan \theta = -3~~~[/latex] and [latex]~90° \lt\ \theta \lt 180°[/latex]
For Problems 67–72, find exact values for the sine, cosine, and tangent of the angle.
[latex]2\cos A + 9 = 8~~~[/latex] and [latex]~90° \lt\ A \lt 180°[/latex]
[latex]25\sin B + 8 = -12~~~[/latex] and [latex]~180° \lt\ B \lt 270°[/latex]
[latex]8\tan \beta + 5 = -11~~~[/latex] and [latex]~90° \lt\ \beta \lt 180°[/latex]
[latex]6(\tan \beta - 4) = -24~~~[/latex] and [latex]~90° \lt\ \beta \lt 270°[/latex]
[latex]\tan^2 C - \dfrac{1}{4} = 0~~~[/latex] and [latex]~0° \lt\ C \lt 180°[/latex]
[latex]4\cos^2 A - \cos A = 0~~~[/latex] and [latex]~00° \lt\ A \lt 180°[/latex]
For Problems 73–76, prove the identity by rewriting tangents in terms of sines and cosines. (These problems involve simplifying complex fractions. See the Algebra Refresher to review this skill.)
[latex]\dfrac{\tan \alpha}{1 + \tan \alpha} = \dfrac{\sin \alpha}{\sin \alpha + \cos \alpha}[/latex]
[latex]\dfrac{1 - \tan u}{1 + \tan u} = \dfrac{\cos u - \sin u}{\cos u + \sin u}[/latex]
[latex]\dfrac{1 + \tan^2 \beta}{1 - \tan^2 \beta} = \dfrac{1}{\cos^2 \beta - \sin^2 \beta}[/latex]
[latex]\tan^2 v - \sin^2 v = \tan^2 v \sin^2 v[/latex]
Prove the Pythagorean identity [latex]\cos^2 \theta + \sin^2 \theta = 1[/latex] by carrying out the following steps. Sketch an angle [latex]\theta[/latex] in standard position and label a point [latex](x,y)[/latex] on the terminal side, at a distance [latex]r[/latex] from the vertex.
Prove the tangent identity [latex]\tan \theta = \dfrac{\sin \theta}{\cos \theta}[/latex] by carrying out the following steps. Sketch an angle [latex]\theta[/latex] in standard position and label a point [latex](x,y)[/latex] on the terminal side, at a distance [latex]r[/latex] from the vertex.