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

Formula Used:

sin²x + cos²x = 1

(a + b)² = a² + b² + 2ab

sin 2x = 2 sin x cos x

tan 2x = \(\frac{(2 tan x) }{(1 - tan²x)}\)

Calculation:

⇒ sin x + cos x = \(\frac{1}{5}\)

Squaring both sides:

⇒ (sin x + cos x)² = (\(\frac{1}{5}\))²

⇒ sin²x + cos²x + 2 sin x cos x = \(\frac{1}{25}\)

⇒ 1 + 2 sin x cos x = \(\frac{1}{25}\)

⇒ 1 + sin 2x = \(\frac{1}{25}\)

⇒ sin 2x = \(\frac{1}{25}\) - 1

⇒ sin 2x = -\(\frac{24}{25}\)

Now, we know that sin²2x + cos²2x = 1

⇒ cos²2x = 1 - sin²2x

⇒ cos²2x = 1 - (-\(\frac{24}{25}\))²

⇒ cos²2x = 1 - \(\frac{576}{625}\)

⇒ cos²2x = \(\frac{(625 - 576) }{ 625}\)

⇒ cos²2x = \(\frac{49}{625}\)

⇒ cos 2x = ±\(\frac{7}{25}\)

Since sin 2x is negative, 2x lies in the third or fourth quadrant.

If 2x is in the third quadrant, cos 2x is negative.

If 2x is in the fourth quadrant, cos 2x is positive.

If cos 2x = -\(\frac{7}{25}\), then tan 2x = sin 2x / cos 2x = (-24/25) / (-7/25) = 24/7

If cos 2x = \(\frac{7}{25}\), then tan 2x = sin 2x / cos 2x = (-24/25) / (7/25) = -24/7

Given the options, tan 2x = \(\frac{24}{7}\)

Hence option 4 is correct