Master the Double Angle Identity in Trigonometry: Simplifying sin^2x with sin(2x) for Optimal Results

sin^2x (Power to Double Angle)

1/2(1-cos2x)

The identity for the sine of double angle states that sin(2x) = 2sin(x)cos(x). We can use this identity to simplify sin^2x in terms of double angle.

Starting with the left side of the identity:

sin^2x = (sin(x))^2

Now, we can use the identity sin(2x) = 2sin(x)cos(x) by setting x = ½(2x) to get:

sin(x) = sin(½(2x)) = ±√[(1-cos(2x))/2]

Substituting this into our expression for sin^2x, we get:

sin^2x = (sin(x))^2 = [(1-cos(2x))/2]^2

Expanding the square on the right side, we get:

sin^2x = [(1-cos(2x))^2]/4

Simplifying the expression on the right side by expanding the square and collecting like terms, we get:

sin^2x = (1/2) – (1/2)cos(2x)

Therefore, sin^2x in terms of double angle is:

sin^2x = (1/2) – (1/2)cos(2x)

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