Which equation represents cos(2x) using the formula involving sine?

The equation that correctly represents cos(2x) using the formula involving sine is derived from a well-known double angle identity. The formula states that the cosine of double an angle can be expressed in terms of sine as follows:

[

\cos(2x) = 1 - 2\sin^2(x)

]

This formula can be obtained by starting from the Pythagorean identity, (\sin^2(x) + \cos^2(x) = 1). By substituting (\cos^2(x)) with (1 - \sin^2(x)) in the identity for (\cos(2x)), we arrive at the expression (1 - 2\sin^2(x)).

This relationship is important in trigonometry as it allows for conversions between sine and cosine, which is often useful in solving problems where one function is easier to work with than the other. Hence, the answer is validated by this foundational identity in trigonometry.