Chapter 6

Use the formula for the cosine of the difference of two angles to solve Exercises \(1-12\) Find the exact value of each expression. $$ \cos \left(45^{\circ}-30^{\circ}\right) $$

Use substitution to determine whether the given \(x\) -value is a solution of the equation. $$ \cos x=\frac{\sqrt{2}}{2}, \quad x=\frac{\pi}{4} $$

use the appropriate formula to express each product as a sum or difference. $$ \sin 6 x \sin 2 x $$

Verify each identity. \(\sin x \sec x=\tan x\)

Use the formula for the cosine of the difference of two angles to solve Exercises \(1-12\) Find the exact value of each expression. $$ \cos \left(120^{\circ}-45^{\circ}\right) $$

Use substitution to determine whether the given \(x\) -value is a solution of the equation. $$ \tan x=\sqrt{3}, \quad x=\frac{\pi}{3} $$

use the appropriate formula to express each product as a sum or difference. $$ \sin 8 x \sin 4 x $$

Verify each identity. \(\cos x \csc x=\cot x\)

Use the formula for the cosine of the difference of two angles to solve Exercises \(1-12\) Find the exact value of each expression. $$ \cos \left(\frac{3 \pi}{4}-\frac{\pi}{6}\right) $$

Use substitution to determine whether the given \(x\) -value is a solution of the equation. $$ \sin x=\frac{\sqrt{3}}{2}, \quad x=\frac{\pi}{6} $$

use the appropriate formula to express each product as a sum or difference. $$ \cos 7 x \cos 3 x $$

Verify each identity. \(\tan (-x) \cos x=-\sin x\)

Use the formula for the cosine of the difference of two angles to solve Exercises \(1-12\) Find the exact value of each expression. $$ \cos \left(\frac{2 \pi}{3}-\frac{\pi}{6}\right) $$

use the appropriate formula to express each product as a sum or difference. $$ \cos 9 x \cos 2 x $$

Use substitution to determine whether the given \(x\) -value is a solution of the equation. $$ \sin x=\frac{\sqrt{2}}{2}, \quad x=\frac{\pi}{3} $$

Verify each identity. \(\cot (-x) \sin x=-\cos x\)

Use the formula for the cosine of the difference of two angles to solve Exercises \(1-12\) In Exercises \(5-8,\) each expression is the right side of the formula for \(\cos (\alpha-\beta)\) with particular values for \(\alpha\) and \(\beta\) a. Identify \(\alpha\) and \(\beta\) in each expression. b. Write the expression as the cosine of an angle. c. Find the exact value of the expression. $$ \cos 50^{\circ} \cos 20^{\circ}+\sin 50^{\circ} \sin 20^{\circ} $$

use the appropriate formula to express each product as a sum or difference. $$ \sin x \cos 2 x $$

Use substitution to determine whether the given \(x\) -value is a solution of the equation. $$ \cos x=-\frac{1}{2}, x=\frac{2 \pi}{3} $$

Verify each identity. \(\tan x \csc x \cos x=1\)

Use the formula for the cosine of the difference of two angles to solve Exercises \(1-12\) In Exercises \(5-8,\) each expression is the right side of the formula for \(\cos (\alpha-\beta)\) with particular values for \(\alpha\) and \(\beta\) a. Identify \(\alpha\) and \(\beta\) in each expression. b. Write the expression as the cosine of an angle. c. Find the exact value of the expression. $$ \cos 50^{\circ} \cos 5^{\circ}+\sin 50^{\circ} \sin 5^{\circ} $$

use the appropriate formula to express each product as a sum or difference. $$ \sin 2 x \cos 3 x $$

Use substitution to determine whether the given \(x\) -value is a solution of the equation. $$ \cos x=-\frac{1}{2}, \quad x=\frac{4 \pi}{3} $$

Verify each identity. \(\cot x \sec x \sin x=1\)

Use the formula for the cosine of the difference of two angles to solve Exercises \(1-12\) In Exercises \(5-8,\) each expression is the right side of the formula for \(\cos (\alpha-\beta)\) with particular values for \(\alpha\) and \(\beta\) a. Identify \(\alpha\) and \(\beta\) in each expression. b. Write the expression as the cosine of an angle. c. Find the exact value of the expression. $$ \cos \frac{5 \pi}{12} \cos \frac{\pi}{12}+\sin \frac{5 \pi}{12} \sin \frac{\pi}{12} $$

use the appropriate formula to express each product as a sum or difference. $$ \cos \frac{3 x}{2} \sin \frac{x}{2} $$

Use substitution to determine whether the given \(x\) -value is a solution of the equation. $$ \tan 2 x=-\frac{\sqrt{3}}{3}, x=\frac{5 \pi}{12} $$

Use the given information to find the exact value of each of the following: a. \(\sin 2 \theta\) b. \(\cos 2 \theta\) c. \(\tan 2 \theta\) \(\sin \theta=\frac{15}{17}, \theta\) lies in quadrant II.

Verify each identity. \(\sec x-\sec x \sin ^{2} x=\cos x\)

Use the formula for the cosine of the difference of two angles to solve Exercises \(1-12\) In Exercises \(5-8,\) each expression is the right side of the formula for \(\cos (\alpha-\beta)\) with particular values for \(\alpha\) and \(\beta\) a. Identify \(\alpha\) and \(\beta\) in each expression. b. Write the expression as the cosine of an angle. c. Find the exact value of the expression. $$ \cos \frac{5 \pi}{18} \cos \frac{\pi}{9}+\sin \frac{5 \pi}{18} \sin \frac{\pi}{9} $$

use the appropriate formula to express each product as a sum or difference. $$ \cos \frac{5 x}{2} \sin \frac{x}{2} $$

Use substitution to determine whether the given \(x\) -value is a solution of the equation. $$ \cos \frac{2 x}{3}=-\frac{1}{2}, x=\pi $$

Use the given information to find the exact value of each of the following: a. \(\sin 2 \theta\) b. \(\cos 2 \theta\) c. \(\tan 2 \theta\) \(\sin \theta=\frac{12}{13}, \theta\) lies in quadrant II.

Verify each identity. \(\csc x-\csc x \cos ^{2} x=\sin x\)

Use the formula for the cosine of the difference of two angles to solve Exercises \(1-12\) Verify each identity. $$ \frac{\cos (\alpha-\beta)}{\cos \alpha \sin \beta}=\tan \alpha+\cot \beta $$

Use substitution to determine whether the given \(x\) -value is a solution of the equation. $$ \cos x=\sin 2 x, \quad x=\frac{\pi}{3} $$

Use the given information to find the exact value of each of the following: a. \(\sin 2 \theta\) b. \(\cos 2 \theta\) c. \(\tan 2 \theta\) \(\cos \theta=\frac{24}{25}, \theta\) lies in quadrant IV.

Verify each identity. \(\cos ^{2} x-\sin ^{2} x=1-2 \sin ^{2} x\)

Use the formula for the cosine of the difference of two angles to solve Exercises \(1-12\) Verify each identity. $$ \frac{\cos (\alpha-\beta)}{\sin \alpha \sin \beta}=\cot \alpha \cot \beta+1 $$

express each sum or difference as a product. If possible, find this product’s exact value. $$ \sin 8 x+\sin 2 x $$

Use substitution to determine whether the given \(x\) -value is a solution of the equation. $$ \cos x+2=\sqrt{3} \sin x, \quad x=\frac{\pi}{6} $$

Use the given information to find the exact value of each of the following: a. \(\sin 2 \theta\) b. \(\cos 2 \theta\) c. \(\tan 2 \theta\) \(\cos \theta=\frac{40}{41}, \theta\) lies in quadrant IV.

Verify each identity. \(\cos ^{2} x-\sin ^{2} x=2 \cos ^{2} x-1\)

Use the formula for the cosine of the difference of two angles to solve Exercises \(1-12\) Verify each identity. $$ \cos \left(x-\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}(\cos x+\sin x) $$

express each sum or difference as a product. If possible, find this product’s exact value. $$ \sin 7 x-\sin 3 x $$

Find all solutions of each equation. $$ \sin x=\frac{\sqrt{3}}{2} $$

Verify each identity. \(\csc \theta-\sin \theta=\cot \theta \cos \theta\)

Use the formula for the cosine of the difference of two angles to solve Exercises \(1-12\) Verify each identity. $$ \cos \left(x-\frac{5 \pi}{4}\right)=-\frac{\sqrt{2}}{2}(\cos x+\sin x) $$

Find all solutions of each equation. $$ \cos x=\frac{\sqrt{3}}{2} $$

Verify each identity. \(\tan \theta+\cot \theta=\sec \theta \csc \theta\)