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

\[ \frac{1 + \cos 2x}{\sin 2x} + 3\left(1 + \tan x \tan\left(\frac{x}{2}\right)\right)\sin x = 4, \]

we will follow these steps:

### Step 1: Simplify \(\frac{1 + \cos 2x}{\sin 2x}\)

Using the double angle identities, we know:

\[ \cos 2x = 2\cos^2 x - 1 \quad \text{and} \quad \sin 2x = 2\sin x \cos x. \]

Thus,

\[ 1 + \cos 2x = 1 + (2\cos^2 x - 1) = 2\cos^2 x. \]

Now substituting this into our equation gives:

\[ \frac{2\cos^2 x}{2\sin x \cos x} = \frac{\cos x}{\sin x} = \cot x. \]

### Step 2: Substitute into the original equation

Now we can rewrite the equation as:

\[ \cot x + 3\left(1 + \tan x \tan\left(\frac{x}{2}\right)\right)\sin x = 4. \]

### Step 3: Simplify the second term

The term \(1 + \tan x \tan\left(\frac{x}{2}\right)\) can be rewritten using the identity for \(\tan\left(\frac{x}{2}\right)\):

\[ \tan\left(\frac{x}{2}\right) = \frac{\sin x}{1 + \cos x}. \]

Thus,

\[ \tan x \tan\left(\frac{x}{2}\right) = \tan x \cdot \frac{\sin x}{1 + \cos x} = \frac{\sin x \tan x}{1 + \cos x} = \frac{\sin^2 x}{\cos x(1 + \cos x)}. \]

### Step 4: Substitute back into the equation

Now substituting back into our equation gives:

\[ \cot x + 3\left(1 + \frac{\sin^2 x}{\cos x(1 + \cos x)}\right)\sin x = 4. \]

### Step 5: Solve for \(\tan x\)

Let \(t = \tan x\). Then, we can express \(\cot x\) as \(\frac{1}{t}\) and rewrite the equation:

\[ \frac{1}{t} + 3\left(1 + t\cdot\frac{\sin x}{1 + \cos x}\right)\sin x = 4. \]

This leads to a quadratic equation in \(t\).

### Step 6: Form the quadratic equation

Rearranging gives:

\[ 3t + \frac{1}{t} = 4 - 3\sin x. \]

Multiplying through by \(t\) gives:

\[ 3t^2 - (4 - 3\sin x)t + 1 = 0. \]

### Step 7: Solve the quadratic equation

Using the quadratic formula \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):

\[ t = \frac{(4 - 3\sin x) \pm \sqrt{(4 - 3\sin x)^2 - 12}}{6}. \]

### Step 8: Find the values of \(\tan x\)

This will yield two possible values for \(\tan x\).