Use the trig identity cos2x = cos^2 x - sin^2 x to verify that cos 2x = 2 cos^2 x -1.?

Answer 1

cos2x = cos^2x - sin^2x = cos^2x - (1 - cos^2x) = 2 cos^2x - 1

Answer 2

The 1 is a cos^2 2x + sin^2 2x in disguise ( 1=sin^2 y+ cos^2 y identity). Plug that in for the 1 and the given identity for cos 2x to get:

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

cos^2 x - sin^2 x = 2 cos^2 x - cos^2 2x sin^2 2x

cos^2 x - sin^2 x = cos^2 2x sin^2 2x

Identity proven. Left side = Right side

Answer 3

Use the trig identity cos2x = cos^2 x - sin^2 x to verify that cos 2x = 2 cos^2 x -1.?

cos²(x) + sin²(x) = 1, from the Pythagorean Theorem.
Solving for sin²(x), you get
sin²(x) = 1 - cos²(x)

Substituting into the identity for cos(2x)
cos(2x)=cos²(x) - sin²(x), we get
cos(2x) = cos²(x) – (1 - cos²(x)) = cos²(x) – 1 + cos²(x) =2cos²(x) – 1

You REALLY NEED to learn this stuff. Just the sin(θ+φ) and cos(θ+φ)… or at least how to derive them. You can get the other functions from these two. You can get the ½-angle identities, and the double angle identities too. If you remember sin(-θ)= -sin(θ) and cos(-θ)=cos(θ) you can also derive the identities for the differences of angles. In other words, remembering only 4 or 5 simple things, you can derive a couple of dozen others.

Answer 4

cos2x = cos^2 x - sin^2 x = cos^2 x - (1 - cos^2 x) = 2*cos^2 x - 1, QED

I also used the identity sin^2 x = 1 - cos^2 x, which is derived from the identity sin^2 x + cos^2 x = 1.

Answer 5

using the trig identity :
sin^2 X + cos^2x=1

this is derived from : (opp/hyp)^2 +(adj/hyp)^2 =1 in a right angle triangle
so sin^2x= 1-cos^2x

substitute 1-cos^2x for sin^2x in the original identity
and you have

cos2x= cos^2x - (1-cos^2x) ---> = cos^2x -1 +cos^2x
which now gives cos2x = 2 cos^2x-1

Answer 6

cos2x = cos^2 x - sin^2 x

we know that

cos^2x + sin^2x = 1 or sin^2x = 1-cos^2x

sub this into the first one

cos2x = cos^2x - (1-cos^2x) = 2cos^2x - 1

Answer 7

guy its no longer an identity in spite of the undeniable fact that it is asserted as its a derived identity that's derived from Sin^2x+cos^2x=a million how the respond is basically positioned tan^x=sin^2x/cos^2x on your eq and you get your answer

Answer 8

sin^2x= 1-cos^2x

so cos 2x= cos^2x-(1-cos^2x) = 2cos^2x-1