Choose all the ones that are equal The image shows a mathematical problem focused on evaluating a trigonometric expression. The expression involves tangent and is formatted as follows:\[\frac{\tan \frac{7\pi}{15} - \tan \frac{2\pi}{15}}{1 + \tan \frac{7\pi}{15} \cdot \tan \frac{2\pi}{15}}\]Below the expression, there is a multiple-choice list of possible answers with checkboxes:1. $\sqrt{3}$2. $-\sqrt{3}$3. $\tan \left( \frac{9\pi}{15} \right)$4. $\tan \left( \frac{2\pi}{3} \right)$5. $\frac{-\sqrt{3}}{3}$ (This option is highlighted and checked, indicating it as the selected answer.)6. $\tan \left( \frac{\pi}{3} \right)$7. $\frac{\sqrt{3}}{3}$8. $\tan \left( \frac{5\pi}{15} \right)$9. $\tan \left( \frac{\pi}{6} \right)$10. $\tan \left( \frac{5\pi}{30} \right)$This problem is likely part of a trigonometry unit focused on using tangent addition formulas or double-angle identities to simplify expressions. The goal is to determine the correct simplification among the given options.

Giventan7π15−tan2π151+tan7π15tan2π15

Answered: tan- tan 15 15 1+ tan tan 15 15 O V3 O…

Trigonometry (11th Edition)

11th Edition

ISBN:9780134217437

Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Chapter1: Trigonometric Functions

Section: Chapter Questions

Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.

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Choose all the ones that are equal

The image shows a mathematical problem focused on evaluating a trigonometric expression. The expression involves tangent and is formatted as follows:

\[
\frac{\tan \frac{7\pi}{15} - \tan \frac{2\pi}{15}}{1 + \tan \frac{7\pi}{15} \cdot \tan \frac{2\pi}{15}}
\]

Below the expression, there is a multiple-choice list of possible answers with checkboxes:

1. $\sqrt{3}$
2. $-\sqrt{3}$
3. $\tan \left( \frac{9\pi}{15} \right)$
4. $\tan \left( \frac{2\pi}{3} \right)$
5. $\frac{-\sqrt{3}}{3}$ (This option is highlighted and checked, indicating it as the selected answer.)
6. $\tan \left( \frac{\pi}{3} \right)$
7. $\frac{\sqrt{3}}{3}$
8. $\tan \left( \frac{5\pi}{15} \right)$
9. $\tan \left( \frac{\pi}{6} \right)$
10. $\tan \left( \frac{5\pi}{30} \right)$

This problem is likely part of a trigonometry unit focused on using tangent addition formulas or double-angle identities to simplify expressions. The goal is to determine the correct simplification among the given options.

Expert Solution

Step 1

$\begin{array}{l} Given \\ \frac{\tan \frac{7 π}{15} - \tan \frac{2 π}{15}}{1 + \tan \frac{7 π}{15} \tan \frac{2 π}{15}} \end{array}$

Step 2

$\begin{array}{l} Using Formula \\ tan (A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \\ Here \\ A= \frac{7 π}{15}, B= \frac{2 π}{15} \\ \frac{\tan \frac{7 π}{15} - \tan \frac{2 π}{15}}{1 + \tan \frac{7 π}{15} \tan \frac{2 π}{15}} = \tan (\frac{7 π}{15} - \frac{2 π}{15}) \\ \frac{\tan \frac{7 π}{15} - \tan \frac{2 π}{15}}{1 + \tan \frac{7 π}{15} \tan \frac{2 π}{15}} = \tan (\frac{7 π - 2 π}{15}) \\ \frac{\tan \frac{7 π}{15} - \tan \frac{2 π}{15}}{1 + \tan \frac{7 π}{15} \tan \frac{2 π}{15}} = \tan (\frac{5 π}{15}) \\ \frac{\tan \frac{7 π}{15} - \tan \frac{2 π}{15}}{1 + \tan \frac{7 π}{15} \tan \frac{2 π}{15}} = \tan (\frac{π}{3}) \\ \frac{\tan \frac{7 π}{15} - \tan \frac{2 π}{15}}{1 + \tan \frac{7 π}{15} \tan \frac{2 π}{15}} = \sqrt{3} \end{array}$

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