The number of solution of the equation `2(tanx-sinx)+3(cotx-cosx)+5=0x epsilon[0,2pi]`

Options

A. `2`
B. `3`
C. `4`
D. `5`

Answer

To solve the equation \(2(\tan x - \sin x) + 3(\cot x - \cos x) + 5 = 0\) for \(x \in [0, 2\pi]\), we will follow these steps:

### Step 1: Rewrite the equation
First, we rewrite the equation using the definitions of tangent and cotangent:
\[ 2\left(\frac{\sin x}{\cos x} - \sin x\right) + 3\left(\frac{\cos x}{\sin x} - \cos x\right) + 5 = 0 \]

### Step 2: Simplify the equation
Distributing the terms:
\[ 2\left(\frac{\sin x - \sin x \cos x}{\cos x}\right) + 3\left(\frac{\cos x - \cos x \sin x}{\sin x}\right) + 5 = 0 \]
This can be rewritten as:
\[ \frac{2\sin x(1 - \cos x)}{\cos x} + \frac{3\cos x(1 - \sin x)}{\sin x} + 5 = 0 \]

### Step 3: Find a common denominator
To combine the fractions, we find a common denominator, which is \(\sin x \cos x\):
\[ \frac{2\sin^2 x(1 - \cos x) + 3\cos^2 x(1 - \sin x) + 5\sin x \cos x}{\sin x \cos x} = 0 \]

### Step 4: Set the numerator to zero
Since the denominator cannot be zero, we set the numerator equal to zero:
\[ 2\sin^2 x(1 - \cos x) + 3\cos^2 x(1 - \sin x) + 5\sin x \cos x = 0 \]

### Step 5: Analyze the equation
This equation is complex, so we will analyze it for solutions. We can substitute \(y = \sin 2x\) since \( \sin 2x = 2 \sin x \cos x\), which simplifies our analysis.

### Step 6: Solve for \(x\)
The solutions for \(2x\) can be found from:
\[ \sin 2x = 0 \]
This gives:
\[ 2x = n\pi \implies x = \frac{n\pi}{2} \]
for \(n = 0, 1, 2, 3, 4\).

### Step 7: Find valid \(n\) values
Now we find valid \(n\) values such that \(x \in [0, 2\pi]\):
- For \(n = 0\): \(x = 0\)
- For \(n = 1\): \(x = \frac{\pi}{2}\)
- For \(n = 2\): \(x = \pi\)
- For \(n = 3\): \(x = \frac{3\pi}{2}\)
- For \(n = 4\): \(x = 2\pi\)

### Step 8: Count the solutions
The valid solutions in the interval \( [0, 2\pi] \) are:
1. \(x = 0\)
2. \(x = \frac{\pi}{2}\)
3. \(x = \pi\)
4. \(x = \frac{3\pi}{2}\)
5. \(x = 2\pi\)

Thus, there are **5 solutions**.

### Final Answer
The number of solutions of the equation is **5**.