Derivative of cotan^2x-cos^2x-cotan^2x

1. Differentiate $- x^{2} n \cot{\left(a \right)} + \left(x^{2} n \cot{\left(a \right)} - \cos^{2}{\left(x \right)}\right)$ term by term:

   1. Differentiate $x^{2} n \cot{\left(a \right)} - \cos^{2}{\left(x \right)}$ term by term:

      1. The derivative of a constant times a function is the constant times the derivative of the function.

         1. Apply the power rule: $x^{2}$ goes to $2 x$

         So, the result is: $2 n x \cot{\left(a \right)}$

      2. The derivative of a constant times a function is the constant times the derivative of the function.

         1. Let $u = \cos{\left(x \right)}$.

         2. Apply the power rule: $u^{2}$ goes to $2 u$

         3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$:

            1. The derivative of cosine is negative sine:

               $$\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$$

            The result of the chain rule is:

            $$- 2 \sin{\left(x \right)} \cos{\left(x \right)}$$

         So, the result is: $2 \sin{\left(x \right)} \cos{\left(x \right)}$

      The result is: $2 n x \cot{\left(a \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}$

   2. The derivative of a constant times a function is the constant times the derivative of the function.

      1. Apply the power rule: $x^{2}$ goes to $2 x$

      So, the result is: $- 2 n x \cot{\left(a \right)}$

   The result is: $2 \sin{\left(x \right)} \cos{\left(x \right)}$

2. Now simplify:

   $$\sin{\left(2 x \right)}$$

The answer is:

$$\sin{\left(2 x \right)}$$