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12 April, 17:36

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

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  1. 12 April, 19:43
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    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)
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