Simplify the trigonometric expression: (sin x + cos x)^2

Answer 1

Start with the FOIL method (first, outside, inside, last). I'll abbreviate Sin and Cos as S and C

(S+C)²=(S+C)(S+C)=S²+S*C+C*S+C² =S²+C²+2SC

=sin²(x)+cos²(x)+2sin(x)cox(x)

then use sin²(x)+cos²(x)=1, and 2sin(x)cox(x)=sin(2x)

(sin(x)+cos(x))²=1+sin(2x)

Answer 2

Treat this just like you would (x + 1)² [which equals x² + 2x + 1]

(sin(x) + cos(x))² = sin²(x) + 2sin(x)cos(x) + cos²(x) now re-order it to be:

sin²(x) + cos²(x) + 2sin(x)cos(x) then use the Pythagorean identity sin²(x) + cos²(x) = 1:to get:

1 + 2sin(x)cos(x)

Then use the double angle identity sin(2x) = 2sin(x)cos(x) to get:

1 + sin(2x)

Answer 3

FOIL:

sin^2 + 2 sin cos + cos^2 =

1 + 2 sin cos

= 2 sin2x+1