Certainly! Let's derive an identity for [tex]\( 2\sin^2(\theta) \)[/tex] using the cosine double-angle identity step-by-step.
The cosine double-angle identity states:
[tex]\[
\cos(2\theta) = 1 - 2\sin^2(\theta)
\][/tex]
Our goal is to derive an identity for [tex]\( 2\sin^2(\theta) \)[/tex] from this equation.
Step 1:
Start with the given cosine double-angle identity:
[tex]\[
\cos(2\theta) = 1 - 2\sin^2(\theta)
\][/tex]
Step 2:
To isolate [tex]\( 2\sin^2(\theta) \)[/tex], subtract [tex]\(\cos(2\theta)\)[/tex] from both sides of the equation:
[tex]\[
\cos(2\theta) - \cos(2\theta) = 1 - 2\sin^2(\theta) - \cos(2\theta)
\][/tex]
Simplifying this, we get:
[tex]\[
0 = 1 - 2\sin^2(\theta) - \cos(2\theta)
\][/tex]
Step 3:
Now, add [tex]\(\cos(2\theta)\)[/tex] to both sides to isolate the [tex]\( 2\sin^2(\theta) \)[/tex] term:
[tex]\[
\cos(2\theta) = 1 - 2\sin^2(\theta)
\][/tex]
[tex]\[
\cos(2\theta) + 2\sin^2(\theta) = 1
\][/tex]
Step 4:
Subtract [tex]\(\cos(2\theta)\)[/tex] from both sides to get [tex]\( 2\sin^2(\theta) \)[/tex] by itself:
[tex]\[
2\sin^2(\theta) = 1 - \cos(2\theta)
\][/tex]
And there you have it! The derived identity for [tex]\( 2\sin^2(\theta) \)[/tex] is:
[tex]\[
2\sin^2(\theta) = 1 - \cos(2\theta)
\][/tex]
