To find [tex]\(\tan 3A\)[/tex] in terms of [tex]\(\tan A\)[/tex], let's analyze the problem step-by-step.
We start with identifying the tangent triple angle formula, which is expressed as:
[tex]\[
\tan(3A) = \frac{3\tan(A) - \tan^3(A)}{1 - 3\tan^2(A)}
\][/tex]
We denote [tex]\(\tan A\)[/tex] as a variable, for simplicity, [tex]\( x = \tan A \)[/tex].
Now, substituting [tex]\( x \)[/tex] into the triple angle identity for [tex]\(\tan\)[/tex], we get:
[tex]\[
\tan(3A) = \frac{3x - x^3}{1 - 3x^2}
\][/tex]
Here’s a detailed breakdown of how the formula is applied:
1. Numerator Evaluation:
- We have [tex]\( 3\tan A \)[/tex] which simplifies directly to [tex]\( 3x \)[/tex].
- We subtract [tex]\( \tan^3 A \)[/tex] which becomes [tex]\( x^3 \)[/tex].
Thus, the numerator of our fraction will be:
[tex]\[
3x - x^3
\][/tex]
2. Denominator Evaluation:
- The denominator starts with 1.
- Then subtract [tex]\( 3\tan^2 A \)[/tex], which translates to [tex]\( 3x^2 \)[/tex].
Therefore, the denominator of our fraction will be:
[tex]\[
1 - 3x^2
\][/tex]
Putting it all together, the expression for [tex]\(\tan(3A)\)[/tex] becomes:
[tex]\[
\tan(3A) = \frac{3x - x^3}{1 - 3x^2}
\][/tex]
Finally, we substitute back [tex]\( x = \tan A \)[/tex] to reach the final form:
[tex]\[
\tan(3A) = \frac{3\tan A - \tan^3 A}{1 - 3\tan^2 A}
\][/tex]
So, the detailed step-by-step solution verifies that:
[tex]\[
\tan(3A) = \frac{3\tan A - \tan^3 A}{1 - 3\tan^2 A}
\][/tex]
And this matches precisely with our expected result.
