Sure, let's break down the steps and their correct order:
To express [tex]\(\tan(2x)\)[/tex] in terms of trigonometric functions of [tex]\(x\)[/tex], we start from different trigonometric identities and rearrange them logically. Let's identify the correct order of steps:
1. Step 3: Start with [tex]\(\tan(2x)\)[/tex], we want to express [tex]\(\tan(2x)\)[/tex] in terms of sine and cosine:
[tex]\[
\tan(2x) = \frac{2 \sin(x) \cos(x)}{\cos^2(x) - \sin^2(x)}
\][/tex]
2. Step 4: Simplify the expression using the identity [tex]\(\sin(2x) = 2\sin(x)\cos(x)\)[/tex] and [tex]\(\cos(2x) = \cos^2(x) - \sin^2(x)\)[/tex]:
[tex]\[
\tan(2x) = \frac{\sin(2x)}{\cos(2x)}
\][/tex]
3. Step 1: Recognize that [tex]\(\tan(x) = \frac{\sin(x)}{\cos(x)}\)[/tex] and substitute [tex]\(\sin(x)\)[/tex] and [tex]\(\cos(x)\)[/tex] back:
[tex]\[
\tan(2x) = \frac{\frac{2\sin(x)}{\cos(x)}}{1 - \frac{\sin^2(x)}{\cos^2(x)}}
\][/tex]
4. Step 2: Simplify using the identity [tex]\(\tan(x) = \frac{\sin(x)}{\cos(x)}\)[/tex]:
[tex]\[
\tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)}
\][/tex]
The detailed, step-by-step solution results in the correct order: [tex]\(3, 4, 1, 5, 2\)[/tex].
So, the correct selection is:
[tex]\[4, 5, 3, 1, 2\][/tex]
