Question

If \[ \sin x + \cos x = \frac{1}{5} \] then \( \tan 2x \) is: 

• A. \( \frac{25}{17} \)
• B. \( \frac{7}{25} \)
• C. \( \sqrt{\frac{25}{7}} \)
• D. \( \frac{24}{7} \)

Hint:

For expressions like \( \sin x + \cos x \), use the transformation \( \sin x + \cos x = \sqrt{2} \sin(x + \pi/4) \).

Answer 1

The Correct Option is D

Step 1: Using identity for \( \sin x + \cos x \).
\[ \sin x + \cos x = \sqrt{2} \sin \left(x + \frac{\pi}{4} \right) \] \[ \Rightarrow \sin \left(x + \frac{\pi}{4} \right) = \frac{1}{5\sqrt{2}} \] Step 2: Finding \( \tan 2x \).
Using the identity: \[ \tan 2x = \frac{2 \sin x \cos x}{\cos^2 x - \sin^2 x} \] Substituting values, we get: \[ \tan 2x = \frac{24}{7} \]