Proving Trigonometric Identity: sin^2(x) = (1 – cos^2(2x)) / 2 using Trigonometric Identities

sin^2(x) = (1 – cos^2(2x)) / 2

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To prove that sin^2(x) = (1 – cos^2(2x)) / 2, we will use the following trigonometric identities.

1. cos(2x) = cos^2(x) – sin^2(x)
2. sin^2(x) + cos^2(x) = 1

Let’s begin with the left-hand side of the equation.

Left-hand side: sin^2(x)

Using identity 2, we know that sin^2(x) = 1 – cos^2(x). We can write this as:

sin^2(x) = 1 – cos^2(x)

Now, we can substitute cos(2x) = cos^2(x) – sin^2(x) into the right-hand side of the equation.

Right-hand side: (1 – cos^2(2x)) / 2

Using identity 1, we can rewrite cos^2(2x) as:

cos^2(2x) = cos^2(x) – sin^2(x)

Substituting this into the right-hand side of the equation, we get:

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

= (1 – cos^2(x) + sin^2(x)) / 2

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

= sin^2(x) / 2

Now, we can substitute sin^2(x) = 1 – cos^2(x) into the last step:

sin^2(x) / 2 = (1 – cos^2(x)) / 2

Putting it all together, we have:

sin^2(x) = (1 – cos^2(2x)) / 2

Therefore, we have proven that sin^2(x) = (1 – cos^2(2x)) / 2 using trigonometric identities.

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