To tackle the given trigonometric equation, let's start by simplifying each component step by step.
The given equation is:
[tex]\[
\frac{\sin x \tan x}{\csc x - \cot x} - \sin x = \tan x
\][/tex]
### Step 1: Express trigonometric functions in terms of [tex]\(\sin x\)[/tex] and [tex]\(\cos x\)[/tex].
Firstly, recall the following trigonometric identities:
[tex]\[
\tan x = \frac{\sin x}{\cos x}
\][/tex]
[tex]\[
\csc x = \frac{1}{\sin x}
\][/tex]
[tex]\[
\cot x = \frac{\cos x}{\sin x}
\][/tex]
### Step 2: Substitute these identities into the equation.
[tex]\[
\frac{\sin x \left( \frac{\sin x}{\cos x} \right)}{\frac{1}{\sin x} - \frac{\cos x}{\sin x}} - \sin x = \frac{\sin x}{\cos x}
\][/tex]
### Step 3: Simplify the expressions.
Let's simplify the denominator first:
[tex]\[
\frac{1}{\sin x} - \frac{\cos x}{\sin x} = \frac{1 - \cos x}{\sin x}
\][/tex]
So, substituting back:
[tex]\[
\frac{\sin x \left( \frac{\sin x}{\cos x} \right)}{\frac{1 - \cos x}{\sin x}} - \sin x
\][/tex]
This simplifies to:
[tex]\[
\frac{\frac{\sin^2 x}{\cos x}}{\frac{1 - \cos x}{\sin x}} - \sin x
\][/tex]
Which can be further simplified as:
[tex]\[
\frac{\sin^3 x}{\cos x (1 - \cos x)} - \sin x
\][/tex]
### Step 4: Equate and simplify the resulting expressions.
Now, our equation looks like:
[tex]\[
\frac{\sin^3 x}{\cos x (1 - \cos x)} - \sin x = \frac{\sin x}{\cos x}
\][/tex]
### Step 5: Multiply through by [tex]\(\cos x (1 - \cos x)\)[/tex] (which should be non-zero), to eliminate the denominators.
[tex]\[
\sin^3 x - \sin x \cos x (1 - \cos x) = \sin x (1 - \cos x)
\][/tex]
### Step 6: Distribute and combine like terms.
[tex]\[
\sin^3 x - \sin x \cos x + \sin x \cos^2 x = \sin x - \sin x \cos x
\][/tex]
Cancel common terms from both sides:
[tex]\[
\sin^3 x + \sin x \cos^2 x = \sin x
\][/tex]
Factor out [tex]\(\sin x\)[/tex] from each term:
[tex]\[
\sin x (\sin^2 x + \cos^2 x) = \sin x
\][/tex]
### Step 7: Simplify using the Pythagorean identity [tex]\(\sin^2 x + \cos^2 x = 1\)[/tex]:
The equation simplifies to:
[tex]\[
\sin x \cdot 1 = \sin x
\][/tex]
[tex]\[
\sin x = \sin x
\][/tex]
This is always true for all [tex]\(x\)[/tex], and hence, the equation simplifies to an identity.
### Conclusion:
The given trigonometric equation simplifies to [tex]\(0 = 0\)[/tex], which suggests there are no specific solutions other than the trivial case where the equation is always true. Considering our methods and reconciling with the conclusion that simplifies to this, we conclude there are no specific solutions or unique values of [tex]\(x\)[/tex].
Therefore, the solution to the given trigonometric equation is:
[tex]\[
0 \quad \text{with no specific solutions for } x.
\][/tex]
