Use the Identities to derive the following?

Question

Using these identities

sin(-x) = -sinx

cos(-x) = cosx

cos(x y) = cosxcosy - sinxsiny

sin(x y) = sinxcosy+cosxsiny

derive

Sin^2x + Cos^2x =1

sin2x = 2 sinXcosX

Answer 1

1.  cos(x-x) = cos(0) = cos(x)cos(-x) - sin(x)sin(-x) =

     cos(x)cos(x) + sin(x)sinx(x) = cos^2(x) + sin^2(x) =  1

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

Answer 2

it's not complicated at all:

from the third identity

cos(x-x)=cosx*cos(-x)-sin^(x)*sin(-x)

cos(0)=cosx*cosx-sin^x*(-sinx)

1=cos^2x+sin^2x

from the fourth identity

sin(x+x)=sinx*cosx+cosx*sinx

sin2x=2sinxcosx

Answer 3

First your identities are wrong. The third and fourth should have (x+y) on the left hand side.

Identity means they are true for all x and y so we can let them be whatever we want.

Let y = x:

sin(x +y) = sinxcosy+cosxsiny

So sin2x = sin (x+x) = sinx cosx +sinx cosx

                              = 2sinxcosx

Let y = -x

1 = cos(x -x) = cosxcos(-x) - sinx sin(-x)

                   = cos x cos x -sinx.(-sin x)

                   = cos^2x +sin^2 x