HW 4.1.2: Simplifying and Verifying Trigonometric Expressions - Weebly

HW 4.1.2: Simplifying and Verifying Trigonometric Expressions

Simplify each of the following to an expression involving a single trig function with no fractions.

1.) csc(t) tan (t)

2.) cos(t) csc(t )

sec (t ) 3.) csc (t )

4.)

sec( ) tan( )

cot (t ) 5.) csc (t )

1- sin2 (t ) 6.) sin2 (t )

sin2 (t ) + cos2 (t )

7.)

cos2 (t )

sec(t ) - cos (t ) 8.) sin (t )

9.) sin x(tan x + cot x)

10.) (1+ sin )(1- sin )

Verify the identity.

11.)

sin2 ( ) 1+ cos( )

= 1- cos(

)

13.)

cos2 ( ) 1+ sin( )

= 1- sin( )

12.)

1+ cot() csc( )

=

sin( ) + cos( )

14.)

1+ tan2 (b) tan2 (b)

=

csc2 (b)

15.) cos(x)(sec(x) + csc(x)) = 1+ cot(x)

16.)

(1+ cos( A))(1- cos( A))

sin ( A)

= sin ( A)

17.)

sin4 ( ) - cos4 ( ) sin ( ) - cos( )

= sin ( ) + cos( )

18.)

1- sin(t) cos2 (t)

+

1

-

1 sin(t

)

=

2 sec2 (t)

19.) (cos x -sin x)2 + (cos x + sin x)2 = 2

20.) sec2 x - tan2 x + tan x = sin x + cos x sec x

Selected Answers: 1.) sec(t) 9.) sec(x)

3.) tan(t)

5.) cos(t)

11.)

sin2 ( ) 1+ cos( )

=

1- cos2 ( ) 1+ cos( )

=

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

1+ cos( )

= 1- cos( )

7.) sec2 (t)

15.)

cos(x)(sec(x) +

csc(x))

=

cos(x)

1 cos(x)

+

1 sin( x )

=1+

cos(x) sin(x)

=

1+

cot(x)

17.)

sin4 ( sin (

) - cos4 ( ) ) - cos( )

=

(sin (

) - cos(

))(sin( ) + cos( ))(sin2 (

sin( ) - cos( )

) + cos2 (

))

= sin(

) + cos(

)

(cos x - sin x)2 + (cos x + sin x)2

( ) ( ) 19.) = cos2 x - 2cos x sin x + sin2 x + cos2 x + 2cos x sin x + sin2 x ( ) = 2 cos2 x + sin2 x = 2

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