How do you prove (sinx - cosx)^2 +(sin x + cosx)^2 = 2?

2 Answers
May 4, 2018

2=2

Explanation:

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

color(red)(sin^2x) - 2 sinx cosx +color(red)(cos^2x) + color(blue)(sin^2x) + 2 sinx cosx +color(blue)(cos^2x) = 2

red terms equal 1
from the Pythagorean theorem
also, blue terms equal 1

So

1 color(green)(- 2 sinx cosx) + 1 color(green)(+2 sinx cosx) = 2

green terms together equal 0

So now you have

1 + 1 = 2

2 = 2

True

May 4, 2018

"see explanation"

Explanation:

"using the "color(blue)"trigonometric identity"

•color(white)(x)sin^2x+cos^2x=1

"consider left side"

"expand each factor using FOIL"

(sinx-cosx)^2=sin^2xcancel(-2cosxsinx)+cos^2x

(sinx+cosx)^2=sin^2xcancel(+2cosxsinx)+cos^2x

"adding the right sides gives"

2sin^2x+2cos^2x

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

=2xx1=2=" right side "rArr"proven"