X x

[Pages:6]Arkansas Tech University MATH 1203: Trigonometry

Dr. Marcel B. Finan

Solution to Assignment 25

3.

4 sin2 x-1 2 sin x+1

=

(2 sin x-1)(2 sin x+1) 2 sin x+1

=

2 sin x - 1.

5.

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

7.

1

1

cos x

sin x

-

=

-

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

cos x - sin x

=

.

sin x cos x

9.

cos x

cos x(1 + sin x)

=

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

cos x(1 + sin x) = 1 - sin2 x

cos x(1 + sin x) =

(cos2 x

1 + sin x =

cos x 1 sin x =+ cos x cos x

= sec x + tan x.

1

12.

sin4 x - cos4 x =(sin2 x + cos2 x)(sin2 x - cos2 x) =(1)(sin2 x - cos2 x) = sin2 x - cos2 x.

15.

sin

x

-

2

+

1 sin

x

sin

x

-

1 sin x

=

sin x sin

- x

2 -

+

1 sin

1

sin x

x

?

sin x sin x

sin2 x - 2 sin x + 1 = sin2 x - 1

(sin x - 1)(sin x - 1) =

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

sin x - 1

=

.

sin x + 1

19.

cos x

cos x 1 - sin x

=

?

1 + sin x 1 + sin x 1 - sin x

cos x(1 - sin x) = 1 - sin2 x

cos x(1 - sin x)

=

cos2 x

1 - sin x =

cos x

1 sin x =-

cos x cos x

= sec x - tan x.

2

21.

cot x + tan x sec x

=

cos x sin x

+

1

sin x cos x

cos x

=

cos x sin x

+

1

sin x cos x

cos x

?

sin x cos x sin x cos x

cos2 x + sin2 x =

sin x

1 = = csc x.

sin x

24.

2 sin x cot x + sin x - 4 cot x - 2 sin x(2 cot x + 1) - 2(2 cot x + 1)

=

2 cot x + 1

2 cot x + 1

(2 cot x + 1)(sin x - 2)

=

= sin x - 2.

2 cot x + 1

27.

sin2 x - cos2 x = sin2 x - (1 - sin2 x) = sin2 x - 1 + sin2 x = 2 sin2 x - 1.

29.

1

1 cos2 x + sin2 x

sin2 x + cos2 x = sin2 x cos2 x

1 = sin2 x cos2 x

= csc2 x sec2 x.

3

31.

1 sec x - cos x = - cos x

cos x

1 - cos2 x =

cos x

sin2 x =

cos x

sin x

= sin x ?

= sin x tan x.

cos x

34.

1 sin x

+

1 sin x

-

1 cos x

1 cos x

1

=

sin 1

x

sin x

+

1 cos x

-

1 cos x

?

sin x cos x sin x cos x

cos x + sin x =

cos x - sin x

cos x + sin x cos x - sin x

=

?

cos x - sin x cos x - sin x

cos2 x - sin2 x =

cos2 x - 2 sin x cos x - sin x

cos2 x - sin2 x

=

.

1 - 2 sin x cos x

37.

1

1 1 + cos x

=

?

1 - cos x 1 - cos x 1 + cos x

1 + cos x =

1 - cos2 x

1 + cos x = sin2 x .

4

41.

1

1

(1 - cos x) - (1 + cos x)

-

=

1 + cos x 1 - cos x (1 + cos x)(1 - cos x)

1 - cos x - 1 - cos x

=

1 - cos2 x

2 cos x = - sin2 x

cos x 1

=-2

?

= -2 cot x csc x.

sin x sin x

43.

1 sin x

1 sin x

+ -

csc x sin x

=

1 sin x

1 sin x

+ -

csc x sin x

?

sin x sin x

1+1 = 1 - sin2 x

2 =.

cos2 x

46.

sec2

x

-

csc2

x

1 =

cos2

x

-

1 sin2

x

sin2 x - cos2 x = sin2 x cos2 x

sin2 x

cos2 x

- sin x cos x sin x cos x

= sin2 x cos2 x

sin x cos x

=

sin x cos x

-

cos x sin x

sin x cos x

tan x - cot x

=

.

sin x cos x

48.

cos x

+

cot x sin x

cos x =

+

cos x sin x

?

sin x

cot x

cot x

2 cos x = cos x

sin x

=2 sin x.

5

53.

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

-

=

cos x 1 - sin x

cos x(1 - sin x)

1 - sin2 x - cos2 x =

cos x(1 - sin x)

1-1

=

= 0.

cos x(1 - sin x)

65.

Not

an

identity.

If

x

=

4

,

the

left

side

is

2

and

the

right

side

is

1.

67.

Not an identity.

If

x

=

0,

the

left

side

is

3 2

and

the

right

side

is

2+ 2

3.

69. Not an identity. If x = 0, the left side is 1 and the right side is 1.

6

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