cos2x = Check all that apply. A. 1 - 2 sin^2 x, B. sin^2 x - cos^2 x, C. cos^2 x - sin^2 x, D. 2 sin^2 x - 1

cos (2x) = 1 - 2 sin² (x) ⇒ A

cos (2x) = cos² (x) - sin² (x) ⇒ C

Step-by-step explanation:

Lets revise the rule of cosine compound angles

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

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

Let us use the first rule above

∵ cos (2x) = cos (x + x)

∵ cos (x + x) = cos (x) cos (x) - sin (x) sin (x)

∴ cos (x + x) = cos² (x) - sin² (x)

∴ cos (2x) = cos² (x) - sin² (x) ⇒ (1)

Lets use the rule sin² (x) + cos² (x) = 1

∵ sin² (x) + cos² (x) = 1

- Subtract sin² (x) from both sides

∴ cos² (x) = 1 - sin² (x)

- Substitute cos² (x) by 1 - sin² (x) in (1)

∵ cos (2x) = 1 - sin² (x) - sin² (x)

- Add the like terms in the right hand side

∴ cos (2x) = 1 - 2 sin² (x) ⇒ (2)

∵ sin² (x) + cos² (x) = 1

- Subtract cos² (x) from both sides

∴ sin² (x) = 1 - cos² (x)

- Substitute sin² (x) by 1 - cos² (x) in (1)

∵ cos (2x) = cos² (x) - (1 - cos² (x))

∴ cos (2x) = cos² (x) - 1 + cos² (x)

- Add the like terms in the right hand side

∴ cos (2x) = 2 cos² (x) - 1 ⇒ (3)