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**1. Homework Statement**

Find [tex]\frac{d}{dx} \sin^{-1}(\sin x)[/tex]

**2. Homework Equations**

**3. The Attempt at a Solution**

Now, I know that the above expression simplifies directly to [tex]\frac{d}{dx} x = 1[/tex], but I attempted it the long way. Here it is:

[tex]\frac{d}{dx} \sin^{-1}(\sin x)[/tex]

[tex]= \frac{\cos x}{\sqrt{1 - \sin^2 x}}[/tex]

[tex]= \frac{\cos x}{\sqrt{\cos^2 x}}[/tex]

[tex]= \frac{\cos x}{|\cos x|}[/tex]

[tex]= 1 \mbox{ if } \cos x > 0[/tex]

[tex]= -1 \mbox{ if } \cos x < 0[/tex]

I know that this is incorrect! The answer should be 1. Did I incorrectly use the definition of [tex]\sqrt{x^2} = |x|[/tex]?