Question

If $\cos A = \dfrac{7}{25}$, then the value of $\tan A + \cot A$ is:

• A. $\dfrac{24}{25}$
• B. $\dfrac{25}{24}$
• C. $\dfrac{625}{576}$
• D. $\dfrac{625}{168}$

Hint:

When $\cos A$ is given, always use the Pythagoras theorem to find the remaining sides before calculating trigonometric ratios.

Answer 1

The Correct Option is D

Step 1: Given information.
We are given $\cos A = \dfrac{7}{25}$. In a right-angled triangle, $\cos A = \dfrac{\text{base}}{\text{hypotenuse}}$. So, base $= 7$ and hypotenuse $= 25$.

Step 2: Find the perpendicular using the Pythagoras theorem.
\[ \text{Perpendicular}^2 = \text{Hypotenuse}^2 - \text{Base}^2 \] \[ \text{Perpendicular}^2 = 25^2 - 7^2 = 625 - 49 = 576 \] \[ \text{Perpendicular} = 24 \]
Step 3: Find $\tan A$ and $\cot A$.
\[ \tan A = \frac{\text{Perpendicular}}{\text{Base}} = \frac{24}{7} \] \[ \cot A = \frac{\text{Base}}{\text{Perpendicular}} = \frac{7}{24} \]
Step 4: Calculate $\tan A + \cot A$.
\[ \tan A + \cot A = \frac{24}{7} + \frac{7}{24} \] \[ \tan A + \cot A = \frac{(24^2 + 7^2)}{24 \times 7} = \frac{576 + 49}{168} = \frac{625}{168} \]
Step 5: Conclusion.
Hence, the value of $\tan A + \cot A$ is $\dfrac{625}{168}$.