Precalculus: Graphical, Numerical, Algebraic 8th Edition solutions

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 16, write the expression in terms of sines and cosines only. Express your answer as a single fraction. csc x + sec x

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 16, write the expression in terms of sines and cosines only. Express your answer as a single fraction. tan x + cot x

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 16, write the expression in terms of sines and cosines only. Express your answer as a single fraction. cos x csc x + sin x sec x

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 16, write the expression in terms of sines and cosines only. Express your answer as a single fraction. \(\sin \theta \cot \theta-\cos \theta \tan \theta\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 16, write the expression in terms of sines and cosines only. Express your answer as a single fraction. \(\frac{\sin x}{\csc x}+\frac{\cos x}{\sec x}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 16, write the expression in terms of sines and cosines only. Express your answer as a single fraction. \(\frac{\sec \alpha}{\cos \alpha}-\frac{\sin \alpha}{\sec \alpha \cos ^{2} \alpha}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 712, determine whether or not the equation is an identity. If not, find a single value of x for which the two expressions are not equal. \(\sqrt{x^{2}}=x\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 712, determine whether or not the equation is an identity. If not, find a single value of x for which the two expressions are not equal. \(\sqrt[3]{x^{3}}=x\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 712, determine whether or not the equation is an identity. If not, find a single value of x for which the two expressions are not equal. \(\sqrt{1-\cos ^{2} x}=\sin x\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 712, determine whether or not the equation is an identity. If not, find a single value of x for which the two expressions are not equal. \(\sqrt{\sec ^{2} x-1}=\tan x\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 712, determine whether or not the equation is an identity. If not, find a single value of x for which the two expressions are not equal. \(\ln \frac{1}{x}=-\ln x\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 712, determine whether or not the equation is an identity. If not, find a single value of x for which the two expressions are not equal. \(\ln x^{2}=2 \ln x\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 14, prove the algebraic identity by starting with the LHS expression and supplying a sequence of equivalent expressions that ends with the RHS expression. \(\frac{x^{3}-x^{2}}{x}-(x-1)(x+1)=1-x\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 14, prove the algebraic identity by starting with the LHS expression and supplying a sequence of equivalent expressions that ends with the RHS expression. \(\frac{1}{x}-\frac{1}{2}=\frac{2-x}{2 x}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 14, prove the algebraic identity by starting with the LHS expression and supplying a sequence of equivalent expressions that ends with the RHS expression. \(\frac{x^{2}-4}{x-2}-\frac{x^{2}-9}{x+3}=5\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 14, prove the algebraic identity by starting with the LHS expression and supplying a sequence of equivalent expressions that ends with the RHS expression. (x - 1)(x + 2) - (x + 1)(x - 2) = 2x

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 510, tell whether or not f(x) = sin x is an identity. \(f(x)=\frac{\sin ^{2} x+\cos ^{2} x}{\csc x}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 510, tell whether or not f(x) = sin x is an identity. \(f(x)=\frac{\tan x}{\sec x}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 510, tell whether or not f(x) = sin x is an identity. \(f(x)=\cos x \cdot \cot x\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 510, tell whether or not f(x) = sin x is an identity. \(f(x)=\cos (x-\pi / 2)\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 510, tell whether or not f(x) = sin x is an identity. \(f(x)=\left(\sin ^{3} x\right)\left(1+\cot ^{2} x\right)\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 510, tell whether or not f(x) = sin x is an identity. \(f(x)=\frac{\sin 2 x}{2}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \((\cos x)(\tan x+\sin x \cot x)=\sin x+\cos ^{2} x\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \((\sin x)(\cot x+\cos x \tan x)=\cos x+\sin ^{2} x\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \((1-\tan x)^{2}=\sec ^{2} x-2 \tan x\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \((\cos x-\sin x)^{2}=1-2 \sin x \cos x\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\frac{(1-\cos u)(1+\cos u)}{\cos ^{2} u}=\tan ^{2} u\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\tan x+\sec x=\frac{\cos x}{1-\sin x}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\frac{\cos ^{2} x-1}{\cos x}=-\tan x \sin x\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\frac{\sec ^{2} \theta-1}{\sin \theta}=\frac{\sin \theta}{1-\sin ^{2} \theta}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \((1-\sin \beta)(1+\csc \beta)=1-\sin \beta+\csc \beta-\sin \beta \csc \beta\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\frac{1}{1-\cos x}+\frac{1}{1+\cos x}=2 \csc ^{2} x\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \((\cos t-\sin t)^{2}+(\cos t+\sin t)^{2}=2\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\sin ^{2} \alpha-\cos ^{2} \alpha=1-2 \cos ^{2} \alpha\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\frac{1+\tan ^{2} x}{\sin ^{2} x+\cos ^{2} x}=\sec ^{2} x\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\frac{1}{\tan \beta}+\tan \beta=\sec \beta \csc \beta\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\frac{\cos \beta}{1+\sin \beta}=\frac{1-\sin \beta}{\cos \beta}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\frac{\sec x+1}{\tan x}=\frac{\sin x}{1-\cos x}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\frac{\tan ^{2} x}{\sec x+1}=\frac{1-\cos x}{\cos x}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\frac{\cot v-1}{\cot v+1}=\frac{1-\tan v}{1+\tan v}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\cot ^{2} x-\cos ^{2} x=\cos ^{2} x \cot ^{2} x\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\tan ^{2} \theta-\sin ^{2} \theta=\tan ^{2} \theta \sin ^{2} \theta\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\cos ^{4} x-\sin ^{4} x=\cos ^{2} x-\sin ^{2} x\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\tan ^{4} t+\tan ^{2} t=\sec ^{4} t-\sec ^{2} t\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \((x \sin \alpha+y \cos \alpha)^{2}+(x \cos \alpha-y \sin \alpha)^{2}=x^{2}+y^{2}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\frac{1-\cos \theta}{\sin \theta}=\frac{\sin \theta}{1+\cos \theta}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\frac{\tan x}{\sec x-1}=\frac{\sec x+1}{\tan x}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\frac{\sin t}{1+\cos t}+\frac{1+\cos t}{\sin t}=2 \csc t\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\frac{\sin x-\cos x}{\sin x+\cos x}=\frac{2 \sin ^{2} x-1}{1+2 \sin x \cos x}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\frac{1+\cos x}{1-\cos x}=\frac{\sec x+1}{\sec x-1}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\frac{\sin t}{1-\cos t}+\frac{1+\cos t}{\sin t}=\frac{2(1+\cos t)}{\sin t}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\frac{\sin A \cos B+\cos A \sin B}{\cos A \cos B-\sin A \sin B}=\frac{\tan A+\tan B}{1-\tan A \tan B}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\sin ^{2} x \cos ^{3} x=\left(\sin ^{2} x-\sin ^{4} x\right)(\cos x)\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\sin ^{5} x \cos ^{2} x=\left(\cos ^{2} x-2 \cos ^{4} x+\cos ^{6} x\right)(\sin x)\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\cos ^{5} x=\left(1-2 \sin ^{2} x+\sin ^{4} x\right)(\cos x)\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\sin ^{3} x \cos ^{3} x=\left(\sin ^{3} x-\sin ^{5} x\right)(\cos x)\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\frac{\tan x}{1-\cot x}+\frac{\cot x}{1-\tan x}=1+\sec x \csc x\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\frac{\cos x}{1+\sin x}+\frac{\cos x}{1-\sin x}=2 \sec x\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\frac{2 \tan x}{1-\tan ^{2} x}+\frac{1}{2 \cos ^{2} x-1}=\frac{\cos x+\sin x}{\cos x-\sin x}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\frac{1-3 \cos x-4 \cos ^{2} x}{\sin ^{2} x}=\frac{1-4 \cos x}{1-\cos x}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\cos ^{3} x=\left(1-\sin ^{2} x\right)(\cos x)\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\sec ^{4} x=\left(1+\tan ^{2} x\right)\left(\sec ^{2} x\right)\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1151, prove the identity. \(\sin ^{5} x=\left(1-2 \cos ^{2} x+\cos ^{4} x\right)(\sin x)\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 5257, match the function with an equivalent expression from the following list. Then confirm the match with a proof. (The matching is not one-to-one.) (a) \(\sec ^{2} x \csc ^{2} x\) (b) sec x + tan x (c) \(2 \sec ^{2} x\) (d) tan x sin x

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 5257, match the function with an equivalent expression from the following list. Then confirm the match with a proof. (The matching is not one-to-one.) (a) \(\sec ^{2} x \csc ^{2} x\) (b) sec x + tan x (c) \(2 \sec ^{2} x\) (d) tan x sin x

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 5257, match the function with an equivalent expression from the following list. Then confirm the match with a proof. (The matching is not one-to-one.) (a) \(\sec ^{2} x \csc ^{2} x\) (b) sec x + tan x (c) \(2 \sec ^{2} x\) (d) tan x sin x

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 5257, match the function with an equivalent expression from the following list. Then confirm the match with a proof. (The matching is not one-to-one.) (a) \(\sec ^{2} x \csc ^{2} x\) (b) sec x + tan x (c) \(2 \sec ^{2} x\) (d) tan x sin x

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 5257, match the function with an equivalent expression from the following list. Then confirm the match with a proof. (The matching is not one-to-one.) (a) \(\sec ^{2} x \csc ^{2} x\) (b) sec x + tan x (c) \(2 \sec ^{2} x\) (d) tan x sin x

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 5257, match the function with an equivalent expression from the following list. Then confirm the match with a proof. (The matching is not one-to-one.) (a) \(\sec ^{2} x \csc ^{2} x\) (b) sec x + tan x (c) \(2 \sec ^{2} x\) (d) tan x sin x

Precalculus: Graphical, Numerical, Algebraic 8th Edition
True or False The equation \(\sqrt{x^{2}}=x\) is an identity. Justify your answer.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
True or False The equation \((\sqrt{x})^{2}=x\) is an identity. Justify your answer.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Multiple Choice If f(x) = g(x) is an identity with domain of validity D, which of the following must be true? I. For any x in D, f(x) is defined. II. For any x in D, g(x) is defined. III. For any x in D, f(x) = g(x). (A) None (B) I and II only (C) I and III only (D) III on

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Multiple Choice Which of these is an efficient first step in proving the identity \(\frac{\sin x}{1-\cos x}=\frac{1+\cos x}{\sin x}\)? (A) \(\frac{\sin x}{1-\cos x}=\frac{\cos \left(\frac{\pi}{2}-x\right)}{1-\cos x}\) (B) \(\frac{\sin x}{1-\cos x}=\frac{\sin x}{\sin ^{2} x+\cos ^{2} x-\cos x}

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Multiple Choice Which of the following could be an intermediate expression in a proof of the identity \(\tan \theta+\sec \theta=\frac{\cos \theta}{1-\sin \theta}\) (A) \(\sin \theta+\cos \theta\) (B) \(\tan \theta+\csc \theta\) (C) \(\frac{\sin \theta+1}{\cos \theta}\) (D) \(\frac

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Multiple Choice If f(x) = g(x) is an identity and \(\frac{f(x)}{g(x)}=k\) which of the following must be false? (A) \(g(x) \neq 0\) (B) f(x) = 0 (C) k = 1 (D) f(x) - g(x) = 0 (E) f(x)g(x) > 0

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 6469, identify a simple function that has the same graph. Then confirm your choice with a proof. sin x cot x

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 6469, identify a simple function that has the same graph. Then confirm your choice with a proof. cos x tan x

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 6469, identify a simple function that has the same graph. Then confirm your choice with a proof. \(\frac{\sin x}{\csc x}+\frac{\cos x}{\sec x}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 6469, identify a simple function that has the same graph. Then confirm your choice with a proof. \(\frac{\csc x}{\sin x}-\frac{\cot x \csc x}{\sec x}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 6469, identify a simple function that has the same graph. Then confirm your choice with a proof. \(\frac{\sin x}{\tan x}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 6469, identify a simple function that has the same graph. Then confirm your choice with a proof. \(\left(\sec ^{2} x\right)\left(1-\sin ^{2} x\right)\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Writing to Learn Let \(\theta\) be any number that is in the domain of all six trig functions. Explain why the natural logarithms of all six basic trig functions of \(\theta\) sum to 0.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
If A and B are complementary angles, prove that \(\sin ^{2} A+\sin ^{2} B=1\).

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Group Activity If your class contains 2n students, write the two expressions from n different identities on separate pieces of paper. (If your class contains an odd number of students, invite your teacher to join you for this activity.) You can use the identities from Exercises 1151 in this sect

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 7378, confirm the identity. \(\sqrt{\frac{1-\sin t}{1+\sin t}}=\frac{1-\sin t}{|\cos t|}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 7378, confirm the identity. \(\sqrt{\frac{1+\cos t}{1-\cos t}}=\frac{1+\cos t}{|\sin t|}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 7378, confirm the identity. \(\sin ^{6} x+\cos ^{6} x=1-3 \sin ^{2} x \cos ^{2} x\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 7378, confirm the identity. \(\cos ^{6} x-\sin ^{6} x=\left(\cos ^{2} x-\sin ^{2} x\right)\left(1-\cos ^{2} x \sin ^{2} x\right)\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 7378, confirm the identity. \(\ln |\tan x|=\ln |\sin x|-\ln |\cos x|\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 7378, confirm the identity. \(\ln |\sec \theta+\tan \theta|+\ln |\sec \theta-\tan \theta|=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Writing to Learn Let \(y_{1}=[\sin (x+0.001)-\sin x] / 0.001\) and \(y_{2}=\cos x\). (a) Use graphs and tables to decide whether \(y_{1}=y_{2}\). (b) Find a value for h so that the graph of \(y_{3}=y_{1}-y_{2}\) in \([-2 \pi, 2 \pi]\) by [-h, h] appears to be a sinusoid. Give a convincing

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Hyperbolic Functions The hyperbolic trigonometric functions are defined as follows: \(\sinh x=\frac{e^{x}-e^{-x}}{2} \quad \cosh x=\frac{e^{x}+e^{-x}}{2} \quad \tanh x=\frac{\sinh x}{\cosh x}\) \(\operatorname{csch} x=\frac{1}{\sinh x} \quad \operatorname{sech} x=\frac{1}{\cosh x} \quad \

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Writing to Learn Write a paragraph to explain why \(\cos x=\cos x+\sin (10 \pi x)\) appears to be an identity when the two sides are graphed in a decimal window. Give a convincing argument that it is not an identity.