Sure! Let's simplify the given expression step-by-step using trigonometric identities.
We are given the expression:
[tex]$
\frac{\tan(57^\circ) - \tan(21^\circ)}{1 + \tan(57^\circ) \tan(21^\circ)}
$[/tex]
First, recall the tangent subtraction identity:
[tex]$
\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}
$[/tex]
Here, [tex]\( A = 57^\circ \)[/tex] and [tex]\( B = 21^\circ \)[/tex]. Therefore, by directly using this identity, we have:
[tex]$
\tan(57^\circ - 21^\circ)
$[/tex]
Now, subtract the angles inside the tangent function:
[tex]$
57^\circ - 21^\circ = 36^\circ
$[/tex]
So, the expression can be simplified to:
[tex]$
\tan(36^\circ)
$[/tex]
Thus, the given expression as a single trigonometric function is:
[tex]$
\tan(36^\circ)
$[/tex]
