Happineszs

Lecture Notes

Trigonometric Identities 1 Sample Problems

page 1

Prove each of the following identities. 1. tan x sin x + cos x = sec x 2. 1 1 + tan x = tan x sin x cos x sin x cos2 x = sin3 x + 1 + sin cos = 2 cos 8. 1 2 cos2 x = tan2 x 1 tan2 x + 1 cos x 1 sin x 2 cos2 x

9. sec x + tan x = 10. sin4 x 11. (sin x

3. sin x 4. cos 1 + sin

cos4 x = 1

cos x)2 + (sin x + cos x)2 = 2

cos x 5. 1 sin x sin4 x 6. sin2 x 7. 1

cos x = 2 tan x 1 + sin x

sin2 x + 4 sin x + 3 3 + sin x 12. = 2x cos 1 sin x 13. cos x 1 sin x tan x = sec x 1 + sin x cos2 x

cos4 x =1 cos2 x

cos x sin x = cos x 1 + sin x

14. tan2 x + 1 + tan x sec x =

Practice Problems

Prove each of the following identities. 1. tan x + 2. cos x 1 = 1 + sin x cos x 1 = 2 tan x sec x 1 + sin x 7. cot x 1 1 tan x = cot x + 1 1 + tan x sin x cos x

1

1 sin x

sin3 x + cos3 x 8. =1 sin x + cos x 9. 10. 1 + sin x 1 sin x

1 + tan2 x 1 3. = 1 tan2 x cos2 x sin2 x 4. tan2 x 5. 6. 1 sin2 x = tan2 x sin2 x

1 sin x = 4 tan x sec x 1 + sin x

tan x + tan y = tan x tan y cot x + cot y tan x) (cos x cot x) = (sin x 1) (cos x 1)

cos x sin x + = 2 csc x sin x 1 cos x

11. (sin x 12.

sec x 1 1 cos x = sec x + 1 1 + cos x

1 + tan x cos x + sin x = 1 tan x cos x sin x
Last revised: March 16, 2011

c copyright Hidegkuti, Powell, 2009

Lecture Notes

Trigonometric Identities 1 Sample Problems - Solutions

page 2

Prove each of the following identities. 1. tan x sin x + cos x = sec x Solution: We will only use the fact that sin2 x + cos2 x = 1 for all values of x. LHS = tan x sin x + cos x = sin2 x sin x sin x + cos x = + cos x cos x cos x sin2 x cos2 x sin2 x + cos2 x 1 = + = = = RHS cos x cos x cos x cos x

2.

1 1 + tan x = tan x sin x cos x Solution: We will only use the fact that sin2 x + cos2 x = 1 for all values of x. LHS = 1 cos x sin x cos2 x + sin2 x 1 + tan x = + = = = RHS tan x sin x cos x sin x cos x sin x cos x

3. sin x sin x cos2 x =