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Chapter 4: Problem 112

Determine whether the statement is true or false. Justify your answer. $$\arctan x=\frac{\arcsin x}{\arccos x}$$

Short Answer

Expert verified
The statement \(\arctan x = \frac{\arcsin x}{\arccos x}\) is false as proven by testing specific values of \(x\).

Step by step solution

01

Understanding Inverse Trigonometric Functions

Understand inverse trigonometric functions and their relationships. For instance, using the relationship \(\sin^2 x + \cos^2 x = 1\), by dividing by \(\cos^2 x\), we get \(\tan^2 x + 1 = \frac{1}{\cos^2 x}\) or \(\tan x = \frac{\sin x}{\cos x}\). The inverse of this implies \(\arctan x = \frac{\arcsin x}{\arccos x}\).
02

Testing the Statement

Test this statement with a few values. For example, try \(x = 1\) or \(x = 0\). Use a calculator to get the exact trigonometric values. Remember that the domains and ranges of these functions might restrict the values of \(x\) that can be used.
03

Reach a Conclusion

Get concrete values for both sides of the equation for \(x = 1\) and \(x = 0\). If they do not match, conclude that the statement is false. Specify at which values of \(x\) the statement does not hold true.

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