Double Angle Formula for Sine | Simplified Explanation and Derivation of sin^2(2x) = (1 – cos(4x)) / 2

sin^2x (Power to Double Angle)

The trigonometric identity for the double angle of sine, which is often referred to as the power of double angle, is given as:

sin^2(2x) = (1 – cos(4x)) / 2

This formula provides a way to express the square of the sine of a double angle in terms of the cosine of a quadruple angle

The trigonometric identity for the double angle of sine, which is often referred to as the power of double angle, is given as:

sin^2(2x) = (1 – cos(4x)) / 2

This formula provides a way to express the square of the sine of a double angle in terms of the cosine of a quadruple angle.

To understand how this identity is derived, let’s start with the double angle formula for sine:

sin(2x) = 2sin(x)cos(x)

Now, if we square both sides of this equation, we get:

sin^2(2x) = (2sin(x)cos(x))^2
= 4sin^2(x)cos^2(x)

Since the Pythagorean identity states that sin^2(x) + cos^2(x) = 1, we can rewrite this equation as:

sin^2(2x) = 4sin^2(x)(1 – sin^2(x))

Now, let’s substitute sin^2(x) with (1 – cos^2(x)) based on the Pythagorean identity:

sin^2(2x) = 4(1 – cos^2(x))(1 – (1 – cos^2(x)))

Simplifying this further:

sin^2(2x) = 4(1 – cos^2(x))(2cos^2(x))
= 8cos^2(x) – 4cos^4(x)

Finally, we can express this in terms of the cosine of a quadruple angle by using the multiple angle formula for cosine, which is given as:

cos(4x) = 8cos^4(x) – 8cos^2(x) + 1

By rearranging the terms, we obtain the final form of the double angle formula for sine:

sin^2(2x) = (1 – cos(4x)) / 2

So, this formula allows us to express the square of the sine of a double angle in terms of the cosine of a quadruple angle.

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