Question

write the expression as the sine, cosine, or tangent of an angle. then find the exact value of the expression.
\\( \frac { \tan \frac { 3 \pi } { 5 } - \tan \frac { 4 \pi } { 15 } } { 1 + \tan \frac { 3 \pi } { 5 } \tan \frac { 4 \pi } { 15 } } \\)
write the expression as the sine, cosine, or tangent of a single angle.
\\( \frac { \tan \frac { 3 \pi } { 5 } - \tan \frac { 4 \pi } { 15 } } { 1 + \tan \frac { 3 \pi } { 5 } \tan \frac { 4 \pi } { 15 } } = \tan \frac { \pi } { 3 } \\)
(type an exact answer, using \\( \pi \\) as needed. use integers or fractions for any numbers in the expression.)
find the exact value of the expression.
\\( \frac { \tan \frac { 3 \pi } { 5 } - \tan \frac { 4 \pi } { 15 } } { 1 + \tan \frac { 3 \pi } { 5 } \tan \frac { 4 \pi } { 15 } } = \square \\)
(simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the
expression. rationalize all denominators.)

Explanation:

Step1: Use the tangent subtraction formula

The formula for \(\tan(A - B)=\frac{\tan A-\tan B}{1 + \tan A\tan B}\). Here \(A=\frac{3\pi}{5}\) and \(B = \frac{4\pi}{15}\), so \(\frac{\tan\frac{3\pi}{5}-\tan\frac{4\pi}{15}}{1+\tan\frac{3\pi}{5}\tan\frac{4\pi}{15}}=\tan(\frac{3\pi}{5}-\frac{4\pi}{15})\).
Calculate \(\frac{3\pi}{5}-\frac{4\pi}{15}=\frac{9\pi - 4\pi}{15}=\frac{5\pi}{15}=\frac{\pi}{3}\).

Step2: Find the value of \(\tan\frac{\pi}{3}\)

We know that for an equilateral triangle with side length \(a\), if we consider a \(30 - 60-90\) triangle (by bisecting an equilateral triangle), \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). For \(\theta=\frac{\pi}{3}\) (or \(60^{\circ}\)), \(\tan\frac{\pi}{3}=\sqrt{3}\).

Answer:

\(\sqrt{3}\)