If `sinx+cosx=1/5,0lexlepi,` then tan x is equal to

Options

A. `-4/3or, -3/4`
B. `3/4`
C. `4/5`
D. none of these

Explanation

To solve the equation \( \sin x + \cos x = \frac{1}{5} \) for \( \tan x \), we can follow these steps:

### Step 1: Square both sides
We start with the equation:
\[ \sin x + \cos x = \frac{1}{5} \]
Squaring both sides gives:
\[ (\sin x + \cos x)^2 = \left(\frac{1}{5}\right)^2 \]
This simplifies to:
\[ \sin^2 x + \cos^2 x + 2 \sin x \cos x = \frac{1}{25} \]

### Step 2: Use the Pythagorean identity
Using the identity \( \sin^2 x + \cos^2 x = 1 \), we can rewrite the equation:
\[ 1 + 2 \sin x \cos x = \frac{1}{25} \]
Subtracting 1 from both sides gives:
\[ 2 \sin x \cos x = \frac{1}{25} - 1 \]
This simplifies to:
\[ 2 \sin x \cos x = \frac{1 - 25}{25} = \frac{-24}{25} \]

### Step 3: Express \( \sin 2x \)
We know that \( \sin 2x = 2 \sin x \cos x \), so we can write:
\[ \sin 2x = \frac{-24}{25} \]

### Step 4: Relate \( \sin 2x \) to \( \tan x \)
Using the double angle formula for sine:
\[ \sin 2x = \frac{2 \tan x}{1 + \tan^2 x} \]
We can set this equal to our previous result:
\[ \frac{2 \tan x}{1 + \tan^2 x} = \frac{-24}{25} \]

### Step 5: Cross-multiply
Cross-multiplying gives:
\[ 2 \tan x \cdot 25 = -24(1 + \tan^2 x) \]
This simplifies to:
\[ 50 \tan x = -24 - 24 \tan^2 x \]

### Step 6: Rearrange into standard form
Rearranging gives:
\[ 24 \tan^2 x + 50 \tan x + 24 = 0 \]

### Step 7: Solve the quadratic equation
We can use the quadratic formula \( \tan x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 24, b = 50, c = 24 \):
\[ \tan x = \frac{-50 \pm \sqrt{50^2 - 4 \cdot 24 \cdot 24}}{2 \cdot 24} \]
Calculating the discriminant:
\[ 50^2 - 4 \cdot 24 \cdot 24 = 2500 - 2304 = 196 \]
Thus:
\[ \tan x = \frac{-50 \pm 14}{48} \]

### Step 8: Calculate the two possible values
Calculating the two possible values:
1. \( \tan x = \frac{-50 + 14}{48} = \frac{-36}{48} = \frac{-3}{4} \)
2. \( \tan x = \frac{-50 - 14}{48} = \frac{-64}{48} = \frac{-4}{3} \)

### Final Result
Thus, the possible values for \( \tan x \) are:
\[ \tan x = \frac{-3}{4} \quad \text{or} \quad \tan x = \frac{-4}{3} \]