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Chapter 5: Problem 43

Use the half-angle formulas to simplify the expression. $$-\sqrt{\frac{1-\cos 8 x}{1+\cos 8 x}}$$

Short Answer

Expert verified
The simplified expression is \(-\tan(4x)\).

Step by step solution

01

Recognize the Structure

The expression \(-\sqrt{\frac{1-\cos 8x}{1+\cos 8x}}\) looks like it is in the form of a half-angle identity.
02

Identify the Identity

The half-angle identity for tangent is given by \(\tan\frac{\theta}{2}=-\sqrt{\frac{1-\cos(\theta)}{1+\cos(\theta)}}\).
03

Apply the Identity

The given expression fits this identity with \(\theta = 8x\). Therefore, applying the half-angle identity for tangent, the simplified expression is \(-\tan(4x)\).

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Most popular questions from this chapter

Use the sum-to-product formulas to rewrite the sum or difference as a product. $$\sin 5 \theta-\sin 3 \theta$$

Find all solutions of the equation in the interval \([0,2 \pi) .\) Use a graphing utility to graph the equation and verify the solutions. $$\frac{\cos 2 x}{\sin 3 x-\sin x}-1=0$$

Use the half-angle formulas to simplify the expression. $$\sqrt{\frac{1+\cos 4 x}{2}}$$

Determine whether the statement is true or false. Justify your answer. Because the sine function is an odd function, for a negative number \(u, \sin 2 u=-2 \sin u \cos u\).

(a) determine the quadrant in which \(u / 2\) lies, and (b) find the exact values of \(\sin (u / 2), \cos (u / 2),\) and \(\tan (u / 2)\) using the half-angle formulas. $$\sin u=5 / 13, \quad \pi / 2
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