Verify the Identity?

Question

Verify the identity (sinx - cosx) ^2 = 1 - sin2x

Please help. I am unsure how to figure out this question!

10 points! :]

Answer 1

LS

= (sin(x) - cos(x))² = sin²(x) -2sin(x)cos(x) + cos²(x)

= 1 - 2sin(x)cos(x); because sin²(x) + cos²(x) = 1

We have the identity:

sin(x + y) = sin(x)cos(y) + cos(x)sin(y)

If we let y = x we get that:

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

Therefore, going back to where we left off:

= 1 - 2sin(x)cos(x); because sin²(x) + cos²(x) = 1

= 1 - sin(2x)

= RS

Hope  this helps!

Answer 2

(sin(x) - cos(x))² = sin²(x) - 2 sin(x) cos(x) + cos²(x)

Using the identities

sin²(x) + cos²(x) = 1

and

sin(2x) = 2 sin(x) cos(x)

the desired result follows.