Learn How To Simplify Trigonometric Expressions Using The Sine Double-Angle Identity

sin 2x

2sinxcosx

The sine of 2x can be expressed as follows:

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

This identity is derived from the double-angle formula for sine, which states that sin(2θ) = 2 sin(θ) cos(θ). In this case, we can substitute 2x for θ to obtain the identity above.

To prove this identity, we can use the following steps:

1. Start with the double-angle formula for sine: sin(2θ) = 2 sin(θ) cos(θ).

2. Substitute 2x for θ: sin(2x) = 2 sin(x) cos(x).

3. Therefore, sin(2x) = 2 sin(x) cos(x).

We can also use this identity to simplify certain trigonometric expressions. For example, consider the expression sin²(x) – cos²(x). We can convert both terms to sine using the Pythagorean identity:

sin²(x) – cos²(x) = sin²(x) – (1 – sin²(x))

= sin²(x) – 1 + sin²(x)

= 2 sin²(x) – 1

Now we can use the identity sin(2x) = 2 sin(x) cos(x) to simplify the expression even further:

2 sin²(x) – 1 = sin²(2x) – 1

This identity allows us to express sin²(x) – cos²(x) in terms of sine squared of double angle minus one.

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