Trig Identities Summer School
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In problems 1 - 6, match the trigonometric expression with one of the following. a. -1 b. cos x c. cot x d. 1 e. -tan x f. sin x 1.
sec x cos x
2.
cot x sin x
3.
tan2x - sec2x
4.
(1 - cos2x)(csc x)
6.
sin[(π/2) - x] cos[(π /2) - x]
5.
sin (-x) cos(-x)
Title: Jul 217:25 AM (1 of 13)
In exercises 7 - 12, match the trigonometric expression with one of the following. a. csc x b. tan x c. sin2x d. sin x tan x e. sec2x f. sec2x + tan2x 7.
sin x sec x
8.
cos2x(sec2x - 1)
9.
sec4x - tan4x
10.
cot x sec x
11.
sec2x - 1 sin2x
12.
cos2[(π/2) - x] cos x
Title: Jul 217:26 AM (2 of 13)
In problems 13 - 24, use the fundamental identities to simplify the expression. 13. tan x csc x 14. sin x(csc x - sin x)
15.
cos x tan x
16.
sec2x(1 - sin2x)
17.
cot x csc x
18.
csc x sec x
Title: Jul 217:27 AM (3 of 13)
19.
sec x sin x tan x
20.
sin(-x) cos x
21.
tan2x sec2x
22.
1 tan2x + 1
23.
cos( - x)sec x
24.
Title: Jul 217:27 AM (4 of 13)
cot( - x)cos x
In problems 25 - 30, factor the expression and use the fundamental identities to simplify. 25. tan2x - tan2x sin2x 26. sec2x tan2x + sec2x
27.
sin2x sec2x - sin2x
28.
tan4x + 2tan2x + 1
29.
1 - 2cos2x + cos4x
30.
sin4x - cos4x
Title: Jul 217:28 AM (5 of 13)
In problems 31 - 34, perform the multiplication and use the fundamental identities to simplify. 31. (sec x + 1)(sec x - 1) 32. (sin x + cos x)2
33.
(cot x + csc x)(cot x - csc x)
Title: Jul 217:28 AM (6 of 13)
34.
(3 - 3sin x)(3 + 3sin x)
In problems 35 - 38, verify the identity. 35. sin x csc x = 1
36.
cot2x(sec2x - 1) = 1
37.
38.
1 sin x
cos x + sin x tan x = sec x
Title: Jul 217:28 AM (7 of 13)
sin x = cos2x sin x
Find the exact value of the trigonometric function given that sin u = -7/25 and cos v = -4/5. (Both u and v are in Quadrant III.) 7. cos(u + v) 8. sec(v - u)
9.
csc(u + v)
Title: Jul 217:29 AM (8 of 13)
10.
sin(v - u)
Verify the identity. 11. sin(x + y)sin(x - y) = sin2x - sin2y
12.
sin(x + y) + sin(x - y) = 2 sin x cos y
13.
sin( + x) = ½ (cos x + sin x)
14.
cos(- x) - sin ( + x) = 0
15.
tan(+ x) = tan x
Title: Jul 217:30 AM (9 of 13)
Verify the identity.
Title: Jul 217:31 AM (10 of 13)
Title: Jul 217:32 AM (11 of 13)
Title: Jul 217:32 AM (12 of 13)
Title: Jul 217:33 AM (13 of 13)
sec x cos x
2.
cot x sin x
3.
tan2x - sec2x
4.
(1 - cos2x)(csc x)
6.
sin[(π/2) - x] cos[(π /2) - x]
5.
sin (-x) cos(-x)
Title: Jul 217:25 AM (1 of 13)
In exercises 7 - 12, match the trigonometric expression with one of the following. a. csc x b. tan x c. sin2x d. sin x tan x e. sec2x f. sec2x + tan2x 7.
sin x sec x
8.
cos2x(sec2x - 1)
9.
sec4x - tan4x
10.
cot x sec x
11.
sec2x - 1 sin2x
12.
cos2[(π/2) - x] cos x
Title: Jul 217:26 AM (2 of 13)
In problems 13 - 24, use the fundamental identities to simplify the expression. 13. tan x csc x 14. sin x(csc x - sin x)
15.
cos x tan x
16.
sec2x(1 - sin2x)
17.
cot x csc x
18.
csc x sec x
Title: Jul 217:27 AM (3 of 13)
19.
sec x sin x tan x
20.
sin(-x) cos x
21.
tan2x sec2x
22.
1 tan2x + 1
23.
cos( - x)sec x
24.
Title: Jul 217:27 AM (4 of 13)
cot( - x)cos x
In problems 25 - 30, factor the expression and use the fundamental identities to simplify. 25. tan2x - tan2x sin2x 26. sec2x tan2x + sec2x
27.
sin2x sec2x - sin2x
28.
tan4x + 2tan2x + 1
29.
1 - 2cos2x + cos4x
30.
sin4x - cos4x
Title: Jul 217:28 AM (5 of 13)
In problems 31 - 34, perform the multiplication and use the fundamental identities to simplify. 31. (sec x + 1)(sec x - 1) 32. (sin x + cos x)2
33.
(cot x + csc x)(cot x - csc x)
Title: Jul 217:28 AM (6 of 13)
34.
(3 - 3sin x)(3 + 3sin x)
In problems 35 - 38, verify the identity. 35. sin x csc x = 1
36.
cot2x(sec2x - 1) = 1
37.
38.
1 sin x
cos x + sin x tan x = sec x
Title: Jul 217:28 AM (7 of 13)
sin x = cos2x sin x
Find the exact value of the trigonometric function given that sin u = -7/25 and cos v = -4/5. (Both u and v are in Quadrant III.) 7. cos(u + v) 8. sec(v - u)
9.
csc(u + v)
Title: Jul 217:29 AM (8 of 13)
10.
sin(v - u)
Verify the identity. 11. sin(x + y)sin(x - y) = sin2x - sin2y
12.
sin(x + y) + sin(x - y) = 2 sin x cos y
13.
sin( + x) = ½ (cos x + sin x)
14.
cos(- x) - sin ( + x) = 0
15.
tan(+ x) = tan x
Title: Jul 217:30 AM (9 of 13)
Verify the identity.
Title: Jul 217:31 AM (10 of 13)
Title: Jul 217:32 AM (11 of 13)
Title: Jul 217:32 AM (12 of 13)
Title: Jul 217:33 AM (13 of 13)
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