Problem 66
Question
Use the half-angle formulas to simplify the expression. $$\sqrt{\frac{1+\cos 4 x}{2}}$$
Step-by-Step Solution
Verified Answer
The simplified expression of \(\sqrt{\frac{1+\cos 4 x}{2}}\) using the half-angle formula is \(\cos (2x)\).
1Step 1: Recall Half-Angle Formula
Recall the half-angle formula for cosine which is \(\cos (\theta/2) = \pm \sqrt{\frac{1+\cos \theta}{2}}\). This will be used to simplify the given expression.
2Step 2: Compare Given Expression with Half-Angle Formula
Notice that the given expression is in the form of the right-hand side of the half-angle identity. With \(\theta = 4x\) we have: \(4x/2 = 2x\) which will be used as the simplified version of the given expression.
3Step 3: Apply Half-Angle Formula
Applying the half-angle formula directly results in: \(\cos (2x)\).
Other exercises in this chapter
Problem 65
Use the properties of logarithms and trigonometric identities to verify the identity. $$\ln |\cot \theta|=\ln |\cos \theta|-\ln |\sin \theta|$$
View solution Problem 65
Solve the multiple-angle equation. $$\sec 4 x=2$$
View solution Problem 66
Verify the identity. $$\cos (x+y)+\cos (x-y)=2 \cos x \cos y$$
View solution Problem 66
Rewrite the expression so that it is not in fractional form. $$\frac{\csc y}{\cot y}$$
View solution