How would you rewrite the expressions using the power-reducing formulas in Trigonometry?

In terms of the first power of the cosine.

1. cos^4(x)
2. sin^2(x)cos^2(x)

On the book, the answer to the first is:
(1/8)(3 + 4cos2x + cos4x)
and the second:
(1/8)(1 - cos4x)

= cos4(x)

= [ cos2x]2

{Note : cos2x = ( 1 + cos 2x) / 2 }

= [ ( 1/2 )( 1 + cos 2x)]2

{Note : (AB)2 = A2B2}

=   ( 1/2 )2( 1 + cos 2x)2

{ Note :  (AB)2 =A2 + B2 +  2AB }

= ( 1/4 ) ( 1 + 2cos2x + cos22x)

{Note : cos2x = ( 1 + cos 2x) / 2  , cos4x  = ( 1/2 )( 1 + cos 4x ) }

= ( 1/4 ) [ 1 + 2cos2x +( 1/2 )( 1 + cos 4x) ]

Simplfy

= ( 1/4 ) [ 1 + 2cos2x +( 1/2 ) + ( 1/2 )cos 4x ]

Simplfy

= ( 1/4 ) +( 1/4 ) 2cos2x +( 1/4 )( 1/2 ) + ( 1/4 )( 1/2 )cos 4x ]

= ( 1/4 ) +( 1/4 ) 2cos2x + ( 1/8 ) + ( 1/8 ) cos 4x

= ( 1/4 ) + ( 1/8 ) +( 1/2 ) cos2x+ ( 1/8 ) cos 4x

= ( 3/8 ) + ( 1/2 ) cos2x+ ( 1/8 ) cos 4x                          LCM in 4 , 8 is 8

Take out common factors.

= ( 1/8 )[ 3 + 4cos 2x + cos 4x ]
Ther fore

cos4(x) = ( 1/8 )[ 3 + 4cos 2x + cos 4x ]

= cos4(x)

= [ cos2x]2

{Note : cos2x = ( 1 + cos 2x) / 2 }

= [ ( 1/2 )( 1 + cos 2x)]2

{Note : (AB)2 = A2B2}

=   ( 1/2 )2( 1 + cos 2x)2

{ Note :  (AB)2 =A2 + B2 +  2AB }

= ( 1/4 ) ( 1 + 2cos2x + cos22x)

{Note : cos2x = ( 1 + cos 2x) / 2  , cos4x  = ( 1/2 )( 1 + cos 4x ) }

= ( 1/4 ) [ 1 + 2cos2x +( 1/2 )( 1 + cos 4x) ]

Simplfy

= ( 1/4 ) [ 1 + 2cos2x +( 1/2 ) + ( 1/2 )cos 4x ]

Simplfy

= ( 1/4 ) +( 1/4 ) 2cos2x +( 1/4 )( 1/2 ) + ( 1/4 )( 1/2 )cos 4x ]

= ( 1/4 ) +( 1/4 ) 2cos2x + ( 1/8 ) + ( 1/8 ) cos 4x

= ( 1/4 ) + ( 1/8 ) +( 1/2 ) cos2x+ ( 1/8 ) cos 4x

= ( 3/8 ) + ( 1/2 ) cos2x+ ( 1/8 ) cos 4x                          LCM in 4 , 8 is 8

Take out common factors.

= ( 1/8 )[ 3 + 4cos 2x + cos 4x ]
Ther fore

cos4(x) = ( 1/8 )[ 3 + 4cos 2x + cos 4x ]

= sin2(x)cos2(x)

= [sin(x)cos(x)]2

Divide , Multiply by 1/4

( 1/4 )4[sin(x)cos(x)]2

Simplfy

( 1/4 )[2sin(x)cos(x)]2

[Note : 2sinAcosA = sin2A]

( 1/4 )[sin2(x)]2

[Note : sin2A = ( 1- cos2A) / 2  ,  sin22A = ( 1- cos4A) / 2 ]

( 1/4 )[( 1- cos4x) / 2 ]

Simplfy

( 1/8 )( 1 - cos4x )

There fore

sin2(x)cos2(x) = ( 1/8 )( 1 - cos4x )

1.

cos4(x) =  [ cos2x]2

Recall double angle trigonometric identities: cos2x = ( 1 + cos 2x) / 2 }

= [(1/2)( 1 + cos 2x)]2

= ( 1/4) ( 1 + cos 2x)2

= ( 1/4 ) ( 1 + 2cos2x + cos22x)

= ( 1/4 ) [ 1 + 2cos2x +( 1/2 )( 1 + cos 4x) ]

= ( 1/4 ) [ 1 + 2cos2x +( 1/2 ) + ( 1/2 )cos 4x ]

= ( 1/4 ) +( 1/4 ) 2cos2x +( 1/4 )( 1/2 ) + ( 1/4 )( 1/2 )cos 4x ]

= ( 1/4 ) +( 1/4 ) 2cos2x + ( 1/8 ) + ( 1/8 ) cos 4x

= ( 1/4 ) + ( 1/8 ) +( 1/2 ) cos2x+ ( 1/8 ) cos 4x

Write the expression with common denominator i.e., 8

= ( 2/8 ) + ( 1/8 ) +( 4/8 ) cos2x+ ( 1/8 ) cos 4x

= ( 3/8 ) + ( 4/8 ) cos2x+ ( 1/8 ) cos 4x

Take out common factors.

= ( 1/8 )[ 3 + 4cos 2x + cos 4x ]

cos4(x) = ( 1/8 )[ 3 + 4cos 2x + cos 4x ]