$\cos(x)+\dfrac{\cot(x)}{\sec(x)}+\tan(x) \overset{?}{=} \cos(x)\cot(x)$

$\cos(x)+\dfrac{\dfrac{\cos(x)}{\sin(x)}}{\dfrac{1 }{\cos(x)}}+\dfrac{\sin(x)}{\cos(x)} \overset{?}{=} \cos(x)\dfrac{\cos(x)}{\sin(x)}$

$\cos(x)+\dfrac{\cos^2(x)}{\sin(x)} + \dfrac{\sin(x)}{\cos(x)} \overset{?}{=} \dfrac{\cos^2(x)}{\sin(x)}$

$\cos(x) + \dfrac{\sin(x)}{\cos(x)} \overset{?}{=} 0$

$\cos^2(x) \overset{?}{=} -\sin(x)$

No. These two expressions are not equal.

If you meant

$\cos(x)+\dfrac{\cot(x)}{\sec(x)+\tan(x)} \overset{?}{=} \cos(x)\cot(x)$

Then I suggest you learn to use parentheses and use the method I followed above.