To find the value of [tex]\(\tan \left( A - \frac{\pi}{4} \right)\)[/tex] given [tex]\(\tan A = -\sqrt{15}\)[/tex], we can use the tangent subtraction formula. The formula for the tangent of the difference of two angles [tex]\(A\)[/tex] and [tex]\(B\)[/tex] is:
[tex]\[
\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}
\][/tex]
In this case, let [tex]\(A\)[/tex] be the angle whose tangent is given and [tex]\(B = \frac{\pi}{4}\)[/tex] (since [tex]\(\tan\left(\frac{\pi}{4}\right) = 1\)[/tex]). Thus, [tex]\(\tan B = 1\)[/tex].
Substituting the given values into the tangent subtraction formula:
[tex]\[
\tan \left( A - \frac{\pi}{4} \right) = \frac{\tan A - \tan \frac{\pi}{4}}{1 + \tan A \tan \frac{\pi}{4}}
\][/tex]
[tex]\[
\tan \left( A - \frac{\pi}{4} \right) = \frac{-\sqrt{15} - 1}{1 + (-\sqrt{15}) \cdot 1}
\][/tex]
Simplify the numerator and the denominator separately:
Numerator:
[tex]\[
-\sqrt{15} - 1
\][/tex]
Denominator:
[tex]\[
1 - \sqrt{15}
\][/tex]
Thus, the expression becomes:
[tex]\[
\tan \left( A - \frac{\pi}{4} \right) = \frac{-\sqrt{15} - 1}{1 - \sqrt{15}}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{-\sqrt{15} - 1}{1 - \sqrt{15}}}
\][/tex]
