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)
