To simplify the expression [tex]\(\frac{\tan 9x - \tan 5x}{1 + \tan 9x \tan 5x}\)[/tex], we will use a known tangent subtraction formula.
The tangent subtraction formula states:
[tex]\[
\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}
\][/tex]
In our given problem, we can identify [tex]\(A\)[/tex] and [tex]\(B\)[/tex] as follows:
- [tex]\(A = 9x\)[/tex]
- [tex]\(B = 5x\)[/tex]
Using the tangent subtraction formula:
[tex]\[
\tan(9x - 5x) = \frac{\tan 9x - \tan 5x}{1 + \tan 9x \tan 5x}
\][/tex]
Simplifying the right-hand side:
[tex]\[
\tan(9x - 5x) = \frac{\tan 9x - \tan 5x}{1 + \tan 9x \tan 5x}
\][/tex]
We know:
[tex]\[
9x - 5x = 4x
\][/tex]
Therefore:
[tex]\[
\tan(4x) = \frac{\tan 9x - \tan 5x}{1 + \tan 9x \tan 5x}
\][/tex]
Thus, the expression [tex]\(\frac{\tan 9x - \tan 5x}{1 + \tan 9x \tan 5x}\)[/tex] simplifies to:
[tex]\[
\tan(4x)
\][/tex]
So, the simplified form of the given expression is:
[tex]\[
\tan(4x)
\][/tex]
