How do I prove an identity?

how do you prove that

#(cos^4 x - sin^4 x)/ (cos x - sin x) = cos x + sin x#

1 Answer
George C.
Jun 17, 2018

See explanation...

Explanation:

The difference of squares identity can be written:

#A^2-B^2=(A-B)(A+B)#

We will also need:

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

Hence:

#cos^4 x - sin^4 x = (cos^2 x)^2 - (sin^2 x)^2#

#color(white)(cos^4 x - sin^4 x) = (cos^2 x - sin^2 x)(cos^2 x + sin^2 x)#

#color(white)(cos^4 x - sin^4 x) = cos^2 x - sin^2 x#

#color(white)(cos^4 x - sin^4 x) = (cos x - sin x)(cos x + sin x)#

So:

#(cos^4 x - sin^4 x)/(cos x - sin x) = (cos^2 x - sin^2 x)/(cos x - sin x)#

#color(white)((cos^4 x - sin^4 x)/(cos x - sin x)) = (color(red)(cancel(color(black)((cos x - sin x))))(cos x + sin x))/color(red)(cancel(color(black)((cos x - sin x))))#

#color(white)((cos^4 x - sin^4 x)/(cos x - sin x)) = cos x + sin x#

Related questions
Impact of this question
297 views around the world
You can reuse this answer
Creative Commons License