Trig Identities Summer School

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)