if cosx - sinx = 1/sqrt(2)... what steps can i take to prove that sin(2x) = 1/2?

Question

I'm having trouble with a high school math question.

Answer 1

if cosx - sinx = 1/sqrt(2)... what steps can i take to prove that sin(2x) = 1/2?

 

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take square of both sides

 

 

1-2sin(x)cos(x)=1/2

 

2sin(x)cos(x)=1/2

 

sin(2x)=1/2

Answer 2

sin(2x) is a double-angle identity.  This identity is the same as  2sinxcosx.

 

 

So now we rearrange the other equation so that it is in terms of 2sinxcosx.

 

cosx - sinx = 1 / √2

 

 

Rationalize the right side of the equation.

 

cosx - sinx = √(2) / 2

 

 

Square both sides of the equation.

 

(cosx - sinx)² = 2 / 4

 

 

Expand the left side of the equation.

 

(cosx - sinx)(cosx - sinx) = 1 / 2

 

cos²x - 2sinxcosx + sin²x = 1 / 2

 

-2sinxcosx + 1 = 1 / 2

 

 

Subtract 1 on both side of the equation.

 

-2sinxcosx = -1 / 2

 

 

Divide both sides of the equation by -1.

 

2sinxcosx = 1 / 2