Sure, let's find \(\tan(2t)\) using the double-angle formula for tangent. Given:
[tex]\[
\pi < t < \frac{3\pi}{2} \quad \text{and} \quad \tan(t) = 7
\][/tex]
We want to find \(\tan(2t)\). The double-angle formula for tangent is:
[tex]\[
\tan(2t) = \frac{2 \tan(t)}{1 - \tan^2(t)}
\][/tex]
Substitute \(\tan(t) = 7\) into the formula:
[tex]\[
\tan(2t) = \frac{2 \cdot 7}{1 - 7^2}
\][/tex]
Calculate the expressions in the numerator and the denominator:
[tex]\[
\tan(2t) = \frac{14}{1 - 49}
\][/tex]
Simplify the denominator:
[tex]\[
\tan(2t) = \frac{14}{-48}
\][/tex]
Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
[tex]\[
\tan(2t) = \frac{14 \div 2}{-48 \div 2} = \frac{7}{-24}
\][/tex]
Thus, the exact value is:
[tex]\[
\tan(2t) = -\frac{7}{24}
\][/tex]
