Question

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

Answer 1

Answer:

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)