Let's go through the detailed, step-by-step solution for deriving the trigonometric identity for [tex]\(\tan(2x)\)[/tex]. We will match each step with the corresponding reason.
\begin{tabular}{|r|c|}
\hline
Statement & Reason \\
\hline
[tex]\(\tan(2x)\)[/tex] & Given \\
\hline
[tex]\(=\tan(x + x)\)[/tex] & 1 \\
\hline
[tex]\(=\frac{\tan(x) + \tan(x)}{1 - \tan(x)\tan(x)}\)[/tex] & 2 \\
\hline
[tex]\(=\frac{2\tan(x)}{1 - \tan^2(x)}\)[/tex] & 3 \\
\hline
\end{tabular}
- Reason 1: Why can we write [tex]\(\tan(2x)\)[/tex] as [tex]\(\tan(x + x)\)[/tex]?
- This is because of the Angle Addition Formula for tangent, which states that [tex]\(\tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}\)[/tex]. In our case, [tex]\(a = x\)[/tex] and [tex]\(b = x\)[/tex].
- Thus, Reason 1 is "Angle addition formula."
- Reason 2: How do we get from [tex]\(\tan(x + x)\)[/tex] to [tex]\(\frac{\tan(x) + \tan(x)}{1 - \tan(x)\tan(x)}\)[/tex]?
- This step involves applying the Quotient Identity for Tangent using the angle addition formula for tangent. So, we replace [tex]\(\tan(x + x)\)[/tex] with [tex]\(\frac{\tan(x) + \tan(x)}{1 - \tan(x)\tan(x)}\)[/tex].
- Thus, Reason 2 is "Quotient identity for tangent."
- Reason 3: How do we simplify [tex]\(\frac{\tan(x) + \tan(x)}{1 - \tan(x)\tan(x)}\)[/tex] to [tex]\(\frac{2\tan(x)}{1 - \tan^2(x)}\)[/tex]?
- This involves Simplifying the Expression: combining the [tex]\(\tan(x)\)[/tex] terms in the numerator and recognizing that [tex]\(\tan(x) + \tan(x) = 2\tan(x)\)[/tex], and simplifying the denominator as is.
- Thus, Reason 3 is "Simplifying the expression."
Putting it all together, the correct reasons for each step are as follows:
Reason 1 is "Angle addition formula."
Reason 2 is "Quotient identity for tangent."
Reason 3 is "Simplifying the expression."
