sin(x) + sin(x)cot(x)^2 = sec(x)

Steps to Solve

1. Rewrite the equation using trigonometric identities
   We start with the equation:
   $$ \sin(x) + \sin(x) \cot^2(x) = \sec(x) $$
   We can use the identity $\cot(x) = \frac{\cos(x)}{\sin(x)}$, hence $\cot^2(x) = \frac{\cos^2(x)}{\sin^2(x)}$.
   Substituting this into the equation gives us:
   $$ \sin(x) + \sin(x) \left(\frac{\cos^2(x)}{\sin^2(x)}\right) = \sec(x) $$

2. Simplify the equation
   Now, simplify the left side:
   $$ \sin(x) + \frac{\sin(x) \cos^2(x)}{\sin^2(x)} = \sec(x) $$
   That simplifies further to:
   $$ \sin(x) + \frac{\cos^2(x)}{\sin(x)} = \sec(x) $$

3. Express sec(x) in terms of sine and cosine
   Using the identity $\sec(x) = \frac{1}{\cos(x)}$, we rewrite the equation:
   $$ \sin(x) + \frac{\cos^2(x)}{\sin(x)} = \frac{1}{\cos(x)} $$

4. Multiply through by $\sin(x) \cos(x)$ to eliminate the fractions
   Multiplying through gives us:
   $$ \sin^2(x) \cos(x) + \cos^2(x) = \sin(x) $$

5. Rearranging the equation
   We can rearrange it into standard form:
   $$ \sin^2(x) \cos(x) - \sin(x) + \cos^2(x) = 0 $$

6. Let’s apply a trigonometric identity
   Using $\sin^2(x) + \cos^2(x) = 1$:
   Now, replace $\cos^2(x)$ with $1 - \sin^2(x)$ in the equation:
   $$ \sin^2(x) \cos(x) - \sin(x) + (1 - \sin^2(x)) = 0 $$

7. Combine like terms
   This simplifies to:
   $$ \sin^2(x) \cos(x) - \sin(x) + 1 - \sin^2(x) = 0 $$
   or
   $$ \sin^2(x)(\cos(x) - 1) - \sin(x) + 1 = 0 $$

8. Solve the equation for x
   This is a quadratic in terms of $\sin(x)$. Let $y = \sin(x)$:
   $$ y^2 (\cos(x) - 1) - y + 1 = 0 $$

You can use the quadratic formula $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find $y$.