Problem 67 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

After applying the half-angle formula the given expression simplifies to: \(- \sqrt{\frac{1-\cos 8x}{1+\cos 8x}} = \tan 4x\).

Step 1: Identify the Trigonometric Identity

Identify the trigonometric identity to be used. For this instance, the half-angle identity \(\tan \frac{\theta}{2}=\sqrt{\frac{1-\cos \theta}{1+\cos \theta}}\) is identified to be used because the expression matches its right hand side.

Step 2: Apply the Identity

Apply the half-angle identity to the expression. Replace θ with 8x in the formula \(\tan \frac{\theta}{2}=\sqrt{\frac{1-\cos \theta}{1+\cos \theta}}\). This will result to \(\tan \frac{8x}{2}=\sqrt{\frac{1-\cos 8x}{1+\cos 8x}}\).

Step 3: Simplify the Expression

Simplify the expression obtained in step 2. The expression \(\tan \frac{8x}{2}=\sqrt{\frac{1-\cos 8x}{1+\cos 8x}}\) simplifies to be \(\tan 4x = - \sqrt{\frac{1-\cos 8x}{1+\cos 8x}}\). We have the negative sign at the front due to the given expression.

Step 4: Determine the Final Identity

The final step is recognizing that \(\tan 4x\) is the simplified form of the expression \(- \sqrt{\frac{1-\cos 8x}{1+\cos 8x}}\). That is, \(- \sqrt{\frac{1-\cos 8x}{1+\cos 8x}} = \tan 4x\). This is the final identity as the given expression cannot be simplified further