Prove the identity: cos^4(x)-sin^4(x)=1-2sin^2(x).with explanation

Question

im confused on this one cause of the 4th power of the sine. does this mean that we put it in the calculator 4 times? and what exactly does it mean by prove the identity- does it mean like make sure both sides equal each other?

Answer 1

cos ^ 4( x ) - sin ^ 4 ( x ) =

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

______________________________________

Remark:

a ^ 2 - b ^ 2 = ( a + b ) * ( a - b )

______________________________________

= [ cos ^ 2 ( x ) + sin ^ 2 ( x ) ] *

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

______________________________________

Remark:

cos ^ 2 ( x ) + sin ^ 2 ( x ) = 1
______________________________________

= 1 * [ cos ^ 2 ( x ) - sin ^ 2 ( x ) ] =

cos ^ 2 ( x ) - sin ^ 2 ( x ) =

______________________________________

Remark:

cos ^ 2 ( x ) = 1 - sin ^ 2 ( x )
______________________________________

= 1 - sin ^ 2 ( x ) - sin ^ 2 ( x ) =

1 - 2 sin ^ 2 ( x )