Prove the identity? (cos(x) - sin(x)) ^2=1-2sin(x)cos(x

2 Answers
Mar 2, 2018

See below.

Explanation:

Expand #(cosx-sinx)^2:#

#(cosx-sinx)^2=(cosx-sinx)(cosx-sinx)#

#(cosx-sinx)(cosx-sinx)=cos^2x-2sinxcosx+sin^2x#

Thus far, we have

#cos^2x-2sinxcosx+sin^2x=1-2sinxcosx#

Recall the identity

#sin^2x+cos^2x=1#

Rearranging a bit, we see this shows up in our expression:

#cos^2x-2sinxcosx+sin^2x=sin^2x+cos^2x-2sinxcosx#

Apply the identity:

#sin^2x+cos^2x-2sinxcosx=1-2sinxcosx#

#1-2sinxcosx=1-2sinxcosx#

Mar 2, 2018

Here's the identity to prove:

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

Here are the identities I used to solve this problem:

#color(blue)(cos^2(x))+color(red)(sin^2(x))=color(purple)1#

Here's the proof. I'll start with the left side and transform it until it looks exactly like this right side.

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

#color(white)(LHS)=(cos(x)-sin(x))(cos(x)-sin(x))#

#color(white)(LHS)=cos^2(x)-sin(x)cos(x)-sin(x)cos(x)+sin^2(x)#

#color(white)(LHS)=cos^2(x)-2sin(x)cos(x)+sin^2(x)#

#color(white)(LHS)=cos^2(x)+sin^2(x)-2sin(x)cos(x)#

#color(white)(LHS)=color(blue)(cos^2(x))+color(red)(sin^2(x))-2sin(x)cos(x)#

#color(white)(LHS)=color(purple)1-2sin(x)cos(x)#

#color(white)(LHS)=RHS#

This is what we had to prove. Hope this helped!