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Chapter 5: Problem 44

Use the half-angle formulas to simplify the expression. $$-\sqrt{\frac{1-\cos (x-1)}{2}}$$

Short Answer

Expert verified
The simplified form of the expression \( -\sqrt{\frac{1-\cos(x-1)}{2}} \) using half-angle formulas is -\( \sin\frac{(x-1)}{2} \)

Step by step solution

01

Identify the half-angle formula

Realize that the given expression is similar to the sine half-angle formula, which is \( \sin\frac{胃}{2} = \pm\sqrt{\frac{1-\cos 胃}{2}} \). Here, \(胃\) would correspond to \(x-1\). Apply this formula to simplify expression.
02

Apply the formula

Upon aligning the expression with the half-angle formula, it became evident that the expression can be written as -\( \sin\frac{(x-1)}{2} \). Notice that the negative sign outside the expression corresponds to the \(-\) in \( \pm\sqrt{\frac{1-\cos 胃}{2}} \)
03

Check the solution

As a final step always ensure to cross-verify the solution. Plugging the \(x\) values in both the given expression and the simplified result would yield the same results.

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