To simplify the given expression (\csc^2x-1)\sec x(\tan^2x)\cos x-\csc^2(-x) , we will first rewrite the trigonometric functions in terms of sine and cosine.
1. Recall the following trigonometric identities:
\csc x = \frac{1}{\sin x}
\sec x = \frac{1}{\cos x}
\tan x = \frac{\sin x}{\cos x}
2. Substitute these identities into the expression:
(\frac{1}{\sin^2x} - 1)(\frac{1}{\cos x})(\frac{\sin^2x}{\cos x})\cos x - \frac{1}{\sin^2(-x)}
3. Simplify the expression further:
(\frac{1-\sin^2x}{\sin^2x})(\frac{\sin^2x}{\cos^2x})\cos x+\frac{1}{\sin^2x}
(\frac{\cos^2x}{\sin^2x})(\frac{\sin^2x}{\cos^2x})\cos x+\frac{1}{\sin^2x}
\cos x+\frac{1}{\sin^2x}
4. Write the simplified expression in terms of sine and cosine:
\cos x+\csc^2x
\boxed{\cos x+\csc^2x}
So, the simplified expression is \cos x+\csc^2x .