Question

If \( \sin x = \dfrac{3}{5} \), then \( \tan x + \cot x = \)

• A. \( \dfrac{12}{5} \)
• B. \( \dfrac{5}{12} \)
• C. \( \dfrac{25}{12} \)
• D. \( \dfrac{12}{25} \)

Hint:

To find \( \tan x + \cot x \), use the identity \( \tan x + \cot x = \dfrac{1}{\sin x \cos x} \).

Answer 1

The Correct Option is C

Step 1: Use the identity for \( \tan x + \cot x \).
We use the identity: \[ \tan x + \cot x = \dfrac{\sin x}{\cos x} + \dfrac{\cos x}{\sin x} = \dfrac{\sin^2 x + \cos^2 x}{\sin x \cos x}. \] Since \( \sin^2 x + \cos^2 x = 1 \), the formula simplifies to: \[ \tan x + \cot x = \dfrac{1}{\sin x \cos x}. \]

Step 2: Find \( \cos x \).
Using the identity \( \sin^2 x + \cos^2 x = 1 \), we can find \( \cos x \): \[ \cos x = \sqrt{1 - \sin^2 x} = \sqrt{1 - \left( \dfrac{3}{5} \right)^2} = \sqrt{1 - \dfrac{9}{25}} = \sqrt{\dfrac{16}{25}} = \dfrac{4}{5}. \]

Step 3: Calculate \( \tan x + \cot x \).
Now, using \( \sin x = \dfrac{3}{5} \) and \( \cos x = \dfrac{4}{5} \), we find: \[ \tan x + \cot x = \dfrac{1}{\sin x \cos x} = \dfrac{1}{\dfrac{3}{5} \times \dfrac{4}{5}} = \dfrac{1}{\dfrac{12}{25}} = \dfrac{25}{12}. \]

Step 4: Conclusion.
Thus, \( \tan x + \cot x = \dfrac{25}{12} \).

Final Answer: \( \dfrac{25}{12} \).