Question

If \(540^\circ<A<630^\circ\) and \(|\cos A| = \frac{5}{13}\), then \( \tan\frac{A}{2} \tan A =\)

• A. \( \frac{18}{5} \)
• B. \( -\frac{8}{5} \)
• C. \( \frac{8}{5} \)
• D. \( -\frac{18}{5} \)

Hint:

For trigonometric identities, remember to consider the quadrant in which the angle lies to correctly assign the signs to sine and cosine values.

Answer 1

The Correct Option is D

Step 1: Recognize that \( A \) is in the third quadrant where both sine and cosine are negative, thus \( \cos A = -\frac{5}{13} \). 
Step 2: Calculate \( \sin A \) using the Pythagorean identity \( \sin^2 A = 1 - \cos^2 A \). \[ \sin^2 A = 1 - \left(-\frac{5}{13}\right)^2 = 1 - \frac{25}{169} = \frac{144}{169} \] \[ \sin A = -\frac{12}{13} \quad \text{(since \( A \) is in the third quadrant)} \] Step 3: Use the identity \( \tan^2 A = \frac{\sin^2 A}{\cos^2 A} \). \[ \tan^2 A = \frac{\left(-\frac{12}{13}\right)^2}{\left(-\frac{5}{13}\right)^2} = \frac{144}{25} \] Step 4: Find \( \tan^4 A + \tan^2 A \). \[ \tan^4 A = \left(\frac{144}{25}\right)^2 = \frac{20736}{625} \] \[ \tan^4 A + \tan^2 A = \frac{20736}{625} + \frac{144}{25} = \frac{20736 + 900}{625} = \frac{21636}{625} = \frac{18}{5} \]