How to Express the Cosine of a Double Angle in Terms of the Cosine of the Original Angle | Deriving the Formulas

cos^2x (Power to Double Angle)

The cosine of a double angle can be expressed in terms of the cosine of the original angle using the double angle formula for cosine, which states that:

cos(2x) = cos^2(x) – sin^2(x)

Here, cos(2x) represents the cosine of the double angle (2x), and sin(x) represents the sine of the original angle (x)

The cosine of a double angle can be expressed in terms of the cosine of the original angle using the double angle formula for cosine, which states that:

cos(2x) = cos^2(x) – sin^2(x)

Here, cos(2x) represents the cosine of the double angle (2x), and sin(x) represents the sine of the original angle (x).

To derive the expression for cos^2(x) in terms of the double angle, we can rearrange the formula as follows:

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

This equation shows that the square of the cosine of an angle (x) is equal to the sum of the cosine of the double angle (2x) and the square of the sine of the original angle (x).

It is important to note that the double angle formula for cosine can also be expressed in terms of the sine of the original angle:

cos(2x) = 1 – 2sin^2(x)

Using this alternate form, we can derive another expression for cos^2(x) in terms of the double angle:

cos^2(x) = cos(2x) + 2sin^2(x) – 1

Both of these formulas are useful for finding the value of cos^2(x) when given the value of the double angle, or vice versa.

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