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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 + tan x = 1

tan x

sin x cos x

3. sin x sin x cos2 x = sin3 x

cos

1 + sin

4. 1 + sin + cos = 2 sec

5. cos x

cos x = 2 tan x

1 sin x 1 + sin x

6. cos2 x = csc x cos x tan x + cot x

sin4 x cos4 x

7. sin2 x

=1 cos2 x

8.

tan2 x tan2 x + 1

= sin2 x

9. 1

sin x =

cos x

cos x 1 + sin x

10. 1

2 cos2

x

=

tan2 tan2

x x

+

1 1

11. tan2 = csc2 tan2 1 12. sec x + tan x = cos x

1 sin x

13. csc sin

cot = 1 tan

14. sin4 x cos4 x = 1 2 cos2 x

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

sin2 x + 4 sin x + 3 3 + sin x

16.

cos2 x

= 1 sin x

cos x

17.

tan x = sec x

1 sin x

18.

tan2 x + 1 + tan x sec x =

1 + sin x cos2 x

c copyright Hidegkuti, Powell, 2009

Last revised: May 8, 2013

Lecture Notes

Trigonometric Identities 1

page 2

Practice Problems

Prove each of the following identities.

1. tan x + cos x = 1 1 + sin x cos x

2. tan2 x + 1 = sec2 x

1

1

3.

= 2 tan x sec x

1 sin x 1 + sin x

4. tan x + cot x = sec x csc x

1 + tan2 x

1

5. 1

tan2 x = cos2 x

sin2 x

6. tan2 x sin2 x = tan2 x sin2 x

1 cos x sin x

7.

+

= 2 csc x

sin x 1 cos x

sec x 1 1 cos x

8.

=

sec x + 1 1 + cos x

9. 1 + cot2 x = csc2 x

10.

csc2 x csc2

x

1

=

cos2

x

11. cot x 1 = 1 tan x cot x + 1 1 + tan x

12. (sin x + cos x) (tan x + cot x) = sec x + csc x

sin3 x + cos3 x

13.

= 1 sin x cos x

sin x + cos x

cos x + 1 csc x

14.

sin3 x

= 1

cos x

1 + sin x 1 sin x

15.

= 4 tan x sec x

1 sin x 1 + sin x

16. csc4 x cot4 x = csc2 x + cot2 x

sin2 x

1 cos x

17. cos2 x + 3 cos x + 2 = 2 + cos x

18. tan x + tan y = tan x tan y cot x + cot y

1 + tan x cos x + sin x

19.

=

1 tan x cos x sin x

20. (sin x tan x) (cos x cot x) = (sin x 1) (cos x 1)

c copyright Hidegkuti, Powell, 2009

Last revised: May 8, 2013

Lecture Notes

Trigonometric Identities 1

page 3

Sample Problems - Solutions

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.

sin x

sin2 x

sin2 x cos2 x

LHS = tan x sin x + cos x =

sin x + cos x =

+ cos x =

+

cos x

cos x

cos x cos x

sin2 x + cos2 x 1

=

=

= RHS

cos x

cos x

1

1

2.

+ 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.

1

cos x sin x cos2 x + sin2 x

1

LHS =

+ tan x =

+

=

=

= RHS

tan x

sin x cos x sin x cos x sin x cos x

3. sin x sin x cos2 x = sin3 x

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

LHS = sin x sin x cos2 x = sin x 1 cos2 x = sin x sin2 x = RHS

4. cos + 1 + sin = 2 sec

1 + sin

cos

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

cos

1 + sin

cos2

(1 + sin )2

cos2 + (1 + sin )2

LHS =

+

=

+

=

1 + sin

cos

(1 + sin ) cos (1 + sin ) cos

(1 + sin ) cos

cos2 + 1 + 2 sin + sin2 cos2 + sin2 + 1 + 2 sin

2 + 2 sin

=

=

=

(1 + sin ) cos

(1 + sin ) cos

(1 + sin ) cos

2 (1 + sin )

2

1

=

=

=2

= 2 sec = RHS

(1 + sin ) cos cos

cos

cos x

cos x

5. 1 sin x 1 + sin x = 2 tan x

Solution: We will start with the left-hand side. First we bring the fractions to the common denominator. Recall that sin2 x + cos2 x = 1 for all values of x.

LHS = cos x

cos x

cos x (1 + sin x)

=

cos x (1 sin x)

1 sin x 1 + sin x (1 sin x) (1 + sin x) (1 sin x) (1 + sin x)

cos x (1 + sin x) cos x (1 sin x) cos x + cos x sin x cos x + cos x sin x 2 sin x cos x

=

=

(1 sin x) (1 + sin x)

1 sin2 x

= cos2 x

2 sin x = cos x = 2 tan x = RHS

c copyright Hidegkuti, Powell, 2009

Last revised: May 8, 2013

Lecture Notes

Trigonometric Identities 1

page 4

6. cos2 x = csc x cos x tan x + cot x

Solution: We will start with the right-hand side. We will re-write everything in terms of sin x and cos x and simplify. We will again run into the Pythagorean identity, sin2 x + cos2 x = 1.

1

1 cos x

cos x

cos x

RHS

=

csc x cos x tan x + cot x

=

cos x sin x sin x + cos x cos x sin x

=

sin x sin2 x

+ sin x cos x

1 cos2 x

sin x cos x

=

sin x sin2 x + cos2 x

sin x cos x

=

sin x 1

sin x cos x

= cos x cos x sin x = cos2 x = cos2 x = LHS

sin x

1

1

sin4 x cos4 x

7. sin2 x

=1 cos2 x

Solution: We can factor the numerator via the di?erence of squares theorem.

sin4 x cos4 x sin2 x 2

LHS = sin2 x

= cos2 x

sin2 x

= sin2 x + cos2 x = 1 = RHS

(cos2 x)2 sin2 x + cos2 x

= cos2 x

sin2 x

sin2 x cos2 x

cos2 x

8.

tan2 x tan2 x + 1

= sin2 x

Solution:

tan2 x LHS = tan2 x + 1 =

sin x 2

sin2 x

sin2 x

cos x sin x 2

+1 cos x

=

cos2 x sin2 x cos2 x + 1

=

cos2 x sin2 x cos2 x cos2 x + cos2 x

sin2 x

sin2 x

=

cos2 x sin2 x + cos2 x

=

cos2 x 1

=

sin2 x cos2 x

cos2 x = sin2 x = RHS 1

cos2 x

cos2 x

1 sin x cos x

9.

=

cos x 1 + sin x

Solution:

1 sin x 1 sin x

1 sin x 1 + sin x (1 sin x) (1 + sin x)

1 sin2 x

LHS =

=

1=

=

=

cos x

cos x

cos x 1 + sin x

cos x (1 + sin x)

cos x (1 + sin x)

cos2 x

cos x

=

=

= RHS

cos x (1 + sin x) 1 + sin x

c copyright Hidegkuti, Powell, 2009

Last revised: May 8, 2013

Lecture Notes

Trigonometric Identities 1

page 5

10. 1

2 cos2

x

=

tan2 x tan2 x

1 +1

Solution: RHS =

= =

sin2 x

sin2 x cos2 x sin2 x cos2 x

tan2 x 1 tan2 x + 1

=

cos2 x sin2 x

1

=

cos2 x sin2 x

cos2 x cos2 x

=

cos2 x sin2 x + cos2 x

cos2 x + 1 cos2 x + cos2 x

cos2 x

sin2 x cos2 x cos2 x

cos2 x sin2 x + cos2 x

=

sin2 x cos2 x sin2 x + cos2 x

=

sin2 x

1

cos2 x

=

sin2 x

1 cos2 x cos2 x = 1 2 cos2 x = LHS

cos2 x

11. tan2 = csc2 tan2 1

RHS = csc2 tan2

1 1 = sin2

sin 2

1 sin2

1

cos

1 = sin2 cos2 1 = cos2 1

1 = cos2

cos2

1 cos2

sin2

sin

cos2 = cos2 = cos2 = cos

2

= tan2 = LHS

12. sec x + tan x = cos x 1 sin x

Solution:

RHS =

cos x

cos x

=

cos x 1=

1 + sin x =

cos x (1 + sin x)

1 sin x 1 sin x

1 sin x 1 + sin x (1 sin x) (1 + sin x)

cos x (1 + sin x) cos x (1 + sin x) 1 + sin x 1 sin x

=

1 sin2 x =

cos2 x

=

=

+

= LHS

cos x cos x cos x

csc 13.

sin

cot =1

tan

Solution: We will start with the left-hand side. We will re-write everything in terms of sin and cos and simplify. We will again run into the Pythagorean identity, sin2 x + cos2 x = 1 for all angles

x.

1

cos

csc LHS =

sin

cot tan

=

sin sin

sin sin

11 =

sin sin

cos cos

1

sin sin = sin2

cos2 sin2

1

cos

1 cos2

sin2 + cos2

= sin2

=

sin2

cos2

sin2

= sin2 = 1 = RHS

14. sin4 x cos4 x = 1 2 cos2 x

Solution:

LHS = sin4 x cos4 x = sin2 x 2 cos2 x 2 = sin2 x + cos2 x sin2 x cos2 x = 1 sin2 x cos2 x = 1 cos2 x cos2 x = 1 2 cos2 x = RHS

c copyright Hidegkuti, Powell, 2009

Last revised: May 8, 2013

Lecture Notes

Trigonometric Identities 1

page 6

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

Solution:

LHS = (sin x cos x)2 + (sin x + cos x)2 = sin2 x + cos2 x 2 sin x cos x + sin2 x + cos2 x + 2 sin x cos x = 2 sin2 x + 2 cos2 x = 2 sin2 x + cos2 x = 2 1 = 2 = RHS

sin2 x + 4 sin x + 3 3 + sin x

16.

cos2 x

= 1 sin x

Solution:

sin2 x + 4 sin x + 3 (sin x + 1) (sin x + 3) (sin x + 1) (sin x + 3) sin x + 3

LHS =

cos2 x

=

1 sin2 x

=

=

= RHS

(1 + sin x) (1 sin x) 1 sin x

cos x

17.

tan x = sec x

1 sin x

Solution:

cos x

cos x sin x cos2 x sin x (1 sin x) cos2 x sin x + sin2 x

LHS =

tan x =

=

=

1 sin x

1 sin x cos x

cos x (1 sin x)

cos x (1 sin x)

cos2 x + sin2 x sin x

1 sin x

1

=

=

=

= RHS

cos x (1 sin x)

cos x (1 sin x) cos x

18.

tan2 x + 1 + tan x sec x =

1 + sin x cos2 x

Solution:

LHS

=

tan2

x

+

1

+

tan x sec x

=

sin2 x cos2 x

+

1

+

sin x cos x

1 sin2 x cos2 x sin x cos x = cos2 x + cos2 x + cos2 x

sin2 x + cos2 x + sin x 1 + sin x

=

cos2 x

= cos2 x = RHS

For more documents like this, visit our page at and click on Lecture Notes. E-mail questions or comments to mhidegkuti@ccc.edu.

c copyright Hidegkuti, Powell, 2009

Last revised: May 8, 2013

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