Question

If \( \sec\theta = \dfrac{13}{12} \) and \( \theta \) lies in the fourth quadrant, then find \[ \tan\theta \times \csc\theta \times \sin\theta \times \cos\theta \]

• A. \( -\dfrac{5}{13} \)
• B. \( \dfrac{144}{169} \)
• C. \( \dfrac{25}{169} \)
• D. \( \dfrac{5}{13} \)

Hint:

Always decide the signs of trigonometric ratios first based on the quadrant before substitution.

Answer 1

The Correct Option is A

Step 1: Find basic trigonometric ratios.
\[ \sec\theta=\frac{13}{12}\Rightarrow \cos\theta=\frac{12}{13} \] Since \( \theta \) is in the fourth quadrant, \[ \sin\theta=-\frac{5}{13},\quad \tan\theta=-\frac{5}{12},\quad \csc\theta=-\frac{13}{5} \]

Step 2: Substitute in the expression.
\[ \tan\theta \cdot \csc\theta \cdot \sin\theta \cdot \cos\theta = \left(-\frac{5}{12}\right)\left(-\frac{13}{5}\right)\left(-\frac{5}{13}\right)\left(\frac{12}{13}\right) \]

Step 3: Simplify.
\[ = -\frac{5}{13} \]

Step 4: Conclusion.
\[ \boxed{-\dfrac{5}{13}} \]