Answer: Perform each indicated operation and simplify the result so that there are no

Match each expression in Column I with its correct factorization in Column II. x2 - y2 II A. 1x + y21x2 - xy + y22 B. 1x + y21x - y2 C. 1x + y22 D. 1x - y21x2 + xy + y22

Match each expression in Column I with its correct factorization in Column II. x3 - y3 II A. 1x + y21x2 - xy + y22 B. 1x + y21x - y2 C. 1x + y22 D. 1x - y21x2 + xy + y22

Match each expression in Column I with its correct factorization in Column II. x3 + y3 II A. 1x + y21x2 - xy + y22 B. 1x + y21x - y2 C. 1x + y22 D. 1x - y21x2 + xy + y22

Match each expression in Column I with its correct factorization in Column II. x2 + 2xy + y2 II A. 1x + y21x2 - xy + y22 B. 1x + y21x - y2 C. 1x + y22 D. 1x - y21x2 + xy + y22

Fill in the blank(s) to correctly complete each fundamental identity sin2 u + cos2 u =

Fill in the blank(s) to correctly complete each fundamental identity tan2 u + 1 =

Fill in the blank(s) to correctly complete each fundamental identity sin1-u2 =

Fill in the blank(s) to correctly complete each fundamental identity sec1-u2 =

Fill in the blank(s) to correctly complete each fundamental identity tan u = 1= sin u

Fill in the blank(s) to correctly complete each fundamental identity cot u = 1= cos u

Perform each indicated operation and simplify the result so that there are no quotients. cot u +1cot u

Perform each indicated operation and simplify the result so that there are no quotients. sec xcsc x + csc xsec x

Perform each indicated operation and simplify the result so that there are no quotients. tan x1cot x + csc x2

Perform each indicated operation and simplify the result so that there are no quotients. cos b1sec b + csc b2

Perform each indicated operation and simplify the result so that there are no quotients. 1csc2 u+1sec2 u

Perform each indicated operation and simplify the result so that there are no quotients. cos xsec x +sin xcsc x

Perform each indicated operation and simplify the result so that there are no quotients. 1sin a - cos a22

Perform each indicated operation and simplify the result so that there are no quotients. 1tan x + cot x22

Perform each indicated operation and simplify the result so that there are no quotients. 11 + sin t22 + cos2 t

Perform each indicated operation and simplify the result so that there are no quotients. 11 + tan u22 - 2 tan u

Perform each indicated operation and simplify the result so that there are no quotients. 11 + cos x - 11 - cos x

Perform each indicated operation and simplify the result so that there are no quotients. 1sin a - 1 - 1sin a + 1

Factor each trigonometric expression. sin2 u - 1

Factor each trigonometric expression. sec2 u - 1

Factor each trigonometric expression. 1sin x + 122 - 1sin x - 122

Factor each trigonometric expression. 1tan x + cot x22 - 1tan x - cot x22

Factor each trigonometric expression. 2 sin2 x + 3 sin x + 1

Factor each trigonometric expression. 4 tan2 b + tan b - 3

Factor each trigonometric expression. cos4 x + 2 cos2 x + 1

Factor each trigonometric expression. cot4 x + 3 cot2 x + 2

Factor each trigonometric expression. sin3 x - cos3 x

Factor each trigonometric expression. sin3 a + cos3 a

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression. tan u cos u

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression. cot a sin a

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression. sec r cos r

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression. cot t tan t

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression. sin b tan bcos b

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression. csc u sec ucot u

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression. sec2 x - 1

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression. csc2 t - 1

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression. sin2 xcos2 x + sin x csc x

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression. 1tan2 a + cot a tan a

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression. 1 - 1csc2 x

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression. 1 - 1sec2 x

Verify that each equation is an identity. See Examples 15. cot ucsc u = cos u

Verify that each equation is an identity. See Examples 15. tan asec a = sin a

Verify that each equation is an identity. See Examples 15. 1 - sin2 bcos b = cos b

Verify that each equation is an identity. See Examples 15. tan2 a + 1sec a = sec a

Verify that each equation is an identity. See Examples 15. cos2 u1tan2 u + 12 = 1

Verify that each equation is an identity. See Examples 15. sin2 b11 + cot2 b2 = 1

Verify that each equation is an identity. See Examples 15. cot u + tan u = sec u csc u

Verify that each equation is an identity. See Examples 15. sin2 a + tan2 a + cos2 a = sec2 a

Verify that each equation is an identity. See Examples 15. cos asec a +sin acsc a = sec2 a - tan2 a

Verify that each equation is an identity. See Examples 15. sin2 ucos u = sec u - cos u

Verify that each equation is an identity. See Examples 15. sin4 u - cos4 u = 2 sin2 u - 1

Verify that each equation is an identity. See Examples 15. sec4 x - sec2 x = tan4 x + tan2 x

Verify that each equation is an identity. See Examples 15. 1 - cos x1 + cos x = 1cot x - csc x22

Verify that each equation is an identity. See Examples 15. 1sec a - tan a22 = 1 - sin a1 + sin a

Verify that each equation is an identity. See Examples 15. cos u + 1tan2 u = cos usec u - 1

Verify that each equation is an identity. See Examples 15. 1sec u - tan u22 + 1sec u csc u - tan u csc u = 2 tan u

Verify that each equation is an identity. See Examples 15. 11 - sin u+11 + sin u = 2 sec2 u

Verify that each equation is an identity. See Examples 15. 1sec a - tan a = sec a + tan a

Verify that each equation is an identity. See Examples 15. cot a + 1cot a - 1 = 1 + tan a1 - tan a

Verify that each equation is an identity. See Examples 15. csc u + cot utan u + sin u = cot u csc u

Verify that each equation is an identity. See Examples 15. cos usin u cot u = 1

Verify that each equation is an identity. See Examples 15. sin2 u11 + cot2 u2 - 1 = 0

Verify that each equation is an identity. See Examples 15. sec4 u - tan4 usec2 u + tan2 u = sec2 u - tan2 u

Verify that each equation is an identity. See Examples 15. sin4 a - cos4 asin2 a - cos2 a = 1

Verify that each equation is an identity. See Examples 15. tan2 t - 1sec2 t = tan t - cot ttan t + cot t

Verify that each equation is an identity. See Examples 15. cot2 t - 11 + cot2 t= 1 - 2 sin2 t

Verify that each equation is an identity. See Examples 15. sin2 a sec2 a + sin2 a csc2 a = sec2 a

Verify that each equation is an identity. See Examples 15. tan2 a sin2 a = tan2 a + cos2 a - 1

Verify that each equation is an identity. See Examples 15. tan x1 + cos x +sin x1 - cos x = cot x + sec x csc x

Verify that each equation is an identity. See Examples 15. sin u1 - cos u - sin u cos u1 + cos u = csc u11 + cos2 u2

Verify that each equation is an identity. See Examples 15. 1 + cos x1 - cos x - 1 - cos x1 + cos x = 4 cot x csc x

Verify that each equation is an identity. See Examples 15. 1 + sin u1 - sin u - 1 - sin u1 + sin u = 4 tan u sec u

Verify that each equation is an identity. See Examples 15. 1 - sin u1 + sin u = sec2 u - 2 sec u tan u + tan2 u

Verify that each equation is an identity. See Examples 15. sin u + cos u = sin u1 - cot u+ cos u1 - tan u

Verify that each equation is an identity. See Examples 15. -1tan a - sec a + -1tan a + sec a = 2 tan a

Verify that each equation is an identity. See Examples 15. 11 + sin x + cos x22 = 211 + sin x211 + cos x2

Verify that each equation is an identity. See Examples 15. 11 - cos2 a211 + cos2 a2 = 2 sin2 a - sin4 a

Verify that each equation is an identity. See Examples 15. 1sec a + csc a21cos a - sin a2 = cot a - tan a

Verify that each equation is an identity. See Examples 15. 1 - cos x1 + cos x = csc2 x - 2 csc x cot x + cot2 x

Verify that each equation is an identity. See Examples 15. 1 - cos u1 + cos u = 2 csc2 u - 2 csc u cot u - 1

Verify that each equation is an identity. See Examples 15. 12 sin x + cos x22 + 12 cos x - sin x22 = 5

Verify that each equation is an identity. See Examples 15. sin2 x11 + cot x2 + cos2 x11 - tan x2 + cot2 x = csc2 x

Verify that each equation is an identity. See Examples 15. sec x - cos x + csc x - sin x - sin x tan x = cos x cot x

Verify that each equation is an identity. See Examples 15. sin3 u + cos3 u = 1cos u + sin u211 - cos u sin u2

Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically. 1sec u + tan u211 - sin u2

Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically. 1csc u + cot u21sec u - 12

Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically. cos u + 1sin u + tan u

Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically. tan u sin u + cos u

Graph the expressions on each side of the equals symbol to determine whether the equation might be an identity. (Note: Use a domain whose length is at least 2p.) If the equation looks like an identity, then verify it algebraically. See Example 1. 2 + 5 cos xsin x = 2 csc x + 5 cot x

Graph the expressions on each side of the equals symbol to determine whether the equation might be an identity. (Note: Use a domain whose length is at least 2p.) If the equation looks like an identity, then verify it algebraically. See Example 1. 1 + cot2 x = sec2 xsec2 x - 1

Graph the expressions on each side of the equals symbol to determine whether the equation might be an identity. (Note: Use a domain whose length is at least 2p.) If the equation looks like an identity, then verify it algebraically. See Example 1. tan x - cot xtan x + cot x = 2 sin2 x

Graph the expressions on each side of the equals symbol to determine whether the equation might be an identity. (Note: Use a domain whose length is at least 2p.) If the equation looks like an identity, then verify it algebraically. See Example 1. 11 + sin x +11 - sin x = sec2 x

By substituting a number for t, show that the equation is not an identity. sin1csc t2 = 1

By substituting a number for t, show that the equation is not an identity. 2cos2 t = cos t

By substituting a number for t, show that the equation is not an identity. csc t = 21 + cot2 t

By substituting a number for t, show that the equation is not an identity. cos t = 21 - sin2 t

Intensity of a Lamp According to Lamberts law, the intensity of light from a single source on a flat surface at point P is given by I = k cos2 u, where k is a constant. (Source: Winter, C., Solar Power Plants, Springer-Verlag.) (a) Write I in terms of the sine function. (b) Why does the maximum value of I occur when u = 0?

Oscillating Spring The distance or displacement y of a weight attached to an oscillating spring from its natural position is modeled by y = 4 cos 2pt, where t is time in seconds. Potential energy is the energy of position and is given by P = ky2, where k is a constant. The weight has the greatest potential energy when the spring is stretched the most. (Source: Weidner, R. and R. Sells, Elementary Classical Physics, Vol. 2, Allyn & Bacon.) (a) Write an expression for P that involves the cosine function. (b) Use a fundamental identity to write P in terms of sin 2pt.

Modeling) Radio Tuners See Example 6. Let the energy stored in the inductor be given by L1t2 = 3 cos2 6,000,000t and let the energy stored in the capacitor be given by C1t2 = 3 sin2 6,000,000t, where t is time in seconds. The total energy E in the circuit is given by E1t2 = L1t2 + C1t2. Graph L, C, and E in the window 30, 10-64 by 3-1, 44, with Xscl = 10-7 and Yscl = 1. Interpret the graph.

Modeling) Radio Tuners See Example 6. Let the energy stored in the inductor be given by L1t2 = 3 cos2 6,000,000t and let the energy stored in the capacitor be given by C1t2 = 3 sin2 6,000,000t, where t is time in seconds. The total energy E in the circuit is given by E1t2 = L1t2 + C1t2. Make a table of values for L, C, and E starting at t = 0, incrementing by 10-7. Interpret the results.

Modeling) Radio Tuners See Example 6. Let the energy stored in the inductor be given by L1t2 = 3 cos2 6,000,000t and let the energy stored in the capacitor be given by C1t2 = 3 sin2 6,000,000t, where t is time in seconds. The total energy E in the circuit is given by E1t2 = L1t2 + C1t2. Use a fundamental identity to derive a simplified expression for E1t2.