If `((1+cos2x))/(sin2x)+3(1+(tanx)tan.(x)/(2))sin x=4` then the value of `tanx` can be equal to

Options

A. 2
B. `(1)/(2)`
C. 3
D. `(1)/(3)`

Answer

To solve the equation \(\frac{1 + \cos 2x}{\sin 2x} + 3\left(1 + \frac{\tan x \tan x}{2}\right)\sin x = 4\), we will follow these steps:

### Step 1: Use Trigonometric Identities
We know the identities:
\[ \sin 2x = 2 \sin x \cos x \]
\[ \cos 2x = \cos^2 x - \sin^2 x = 1 - 2\sin^2 x = 2\cos^2 x - 1 \]
We can also express \(\cos 2x\) and \(\sin 2x\) in terms of \(\tan x\):
\[ \sin 2x = \frac{2\tan x}{1 + \tan^2 x} \]
\[ \cos 2x = \frac{1 - \tan^2 x}{1 + \tan^2 x} \]

### Step 2: Substitute the Identities
Substituting these into the equation:
\[ \frac{1 + \frac{1 - \tan^2 x}{1 + \tan^2 x}}{\frac{2\tan x}{1 + \tan^2 x}} + 3\left(1 + \frac{\tan^2 x}{2}\right)\sin x = 4 \]

### Step 3: Simplify the Left Side
The left side simplifies to:
\[ \frac{(1 + \frac{1 - \tan^2 x}{1 + \tan^2 x}) (1 + \tan^2 x)}{2\tan x} + 3\left(1 + \frac{\tan^2 x}{2}\right)\sin x \]
This can be further simplified to:
\[ \frac{(2 + 0) (1 + \tan^2 x)}{2\tan x} + 3\left(1 + \frac{\tan^2 x}{2}\right)\sin x \]

### Step 4: Set Up the Equation
Now we have:
\[ \frac{2(1 + \tan^2 x)}{2\tan x} + 3\left(1 + \frac{\tan^2 x}{2}\right)\sin x = 4 \]
This simplifies to:
\[ \frac{1 + \tan^2 x}{\tan x} + 3\left(1 + \frac{\tan^2 x}{2}\right)\sin x = 4 \]

### Step 5: Let \(t = \tan x\)
Let \(t = \tan x\):
\[ \frac{1 + t^2}{t} + 3\left(1 + \frac{t^2}{2}\right)\sin x = 4 \]

### Step 6: Solve for \(t\)
From the equation, we can isolate \(t\):
\[ \cot x + 3t = 4 \]
Substituting \(\cot x = \frac{1}{t}\):
\[ \frac{1}{t} + 3t = 4 \]

### Step 7: Multiply Through by \(t\)
Multiply through by \(t\) to eliminate the fraction:
\[ 1 + 3t^2 = 4t \]
Rearranging gives:
\[ 3t^2 - 4t + 1 = 0 \]

### Step 8: Use the Quadratic Formula
Using the quadratic formula \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[ t = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 3 \cdot 1}}{2 \cdot 3} \]
\[ t = \frac{4 \pm \sqrt{16 - 12}}{6} \]
\[ t = \frac{4 \pm 2}{6} \]

### Step 9: Find the Values of \(t\)
Calculating the two possible values:
1. \(t = \frac{6}{6} = 1\)
2. \(t = \frac{2}{6} = \frac{1}{3}\)

Thus, the possible values of \(\tan x\) are \(1\) and \(\frac{1}{3}\).

### Final Answer
The value of \(\tan x\) can be equal to \(1\) or \(\frac{1}{3}\).

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