Answer to a math question (csc^2x-1)secx(tan^2x)cosx-csc^2(-x)

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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 .

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