Simplifying Trigonometric Expressions with Nested Square Roots

We are asked to simplify the trigonometric expression: \(\sqrt{2 + \sqrt{2 +\sqrt {2 + 2 \cos 8\theta}}}\).

This expression involves nested square roots and the cosine function. We can simplify this step-by-step, starting from the innermost square root, by using a relevant trigonometric identity. The key identity here is the double-angle formula for cosine, which can be rearranged as:

• \(\cos 2A = 2 \cos^2 A - 1\)
• Rearranging gives \(1 + \cos 2A = 2 \cos^2 A\)

Let's simplify the expression from the inside out:

Step 1: Simplify the innermost square root

The innermost term is \(\sqrt {2 + 2 \cos 8\theta}\). We can factor out 2:

\(\sqrt {2(1 + \cos 8\theta)}\)

Using the identity \(1 + \cos 2A = 2 \cos^2 A\), let \(2A = 8\theta\). This means \(A = 4\theta\). Substituting this into the identity:

\(1 + \cos 8\theta = 2 \cos^2 4\theta\)

Now substitute this back into the expression under the square root:

\(\sqrt {2(2 \cos^2 4\theta)} = \sqrt{4 \cos^2 4\theta}\)

Taking the square root, we get \(2 |\cos 4\theta|\). In problems of this type, it is often assumed that the angles are within a range where the cosine values are positive during the simplification steps unless otherwise specified. So, we can assume \(|\cos 4\theta| = \cos 4\theta\).

The simplified innermost square root is \(2 \cos 4\theta\).

Step 2: Simplify the next level square root

Now substitute the result from Step 1 back into the original expression:

\(\sqrt{2 + \sqrt{2 + 2 \cos 4\theta}}\)

Focus on the expression under the square root: \(2 + 2 \cos 4\theta\). Again, factor out 2:

\(2(1 + \cos 4\theta)\)

Using the identity \(1 + \cos 2A = 2 \cos^2 A\), let \(2A = 4\theta\). This means \(A = 2\theta\). Substituting this into the identity:

\(1 + \cos 4\theta = 2 \cos^2 2\theta\)

Now substitute this back into the expression under the square root:

\(\sqrt{2(2 \cos^2 2\theta)} = \sqrt{4 \cos^2 2\theta}\)

Taking the square root, we get \(2 |\cos 2\theta|\). Assuming \(|\cos 2\theta| = \cos 2\theta\), the simplified term is \(2 \cos 2\theta\).

Step 3: Simplify the outermost square root

Substitute the result from Step 2 back into the original expression:

\(\sqrt{2 + 2 \cos 2\theta}\)

Focus on the expression under the square root: \(2 + 2 \cos 2\theta\). Factor out 2:

\(2(1 + \cos 2\theta)\)

Using the identity \(1 + \cos 2A = 2 \cos^2 A\), let \(2A = 2\theta\). This means \(A = \theta\). Substituting this into the identity:

\(1 + \cos 2\theta = 2 \cos^2 \theta\)

Now substitute this back into the expression under the square root:

\(\sqrt{2(2 \cos^2 \theta)} = \sqrt{4 \cos^2 \theta}\)

Taking the square root, we get \(2 |\cos \theta|\). Assuming \(|\cos \theta| = \cos \theta\), the final simplified expression is \(2 \cos \theta\).

Therefore, the expression \(\sqrt{2 + \sqrt{2 +\sqrt {2 + 2 \cos 8\theta}}}\) simplifies to \(2 \cos \theta\).

Comparing this result with the given options:

• Option 1: 2 sin \(\theta\)
• Option 2: 2 cos \(\theta\)
• Option 3: sin 2\(\theta\)
• Option 4: cos 2\(\theta\)

The simplified expression matches Option 2.

Revision Table: Trigonometric Identities Used

Additional Information: Double Angle and Half Angle Formulas

The trigonometric identity \(1 + \cos 2A = 2 \cos^2 A\) is derived from the double angle formula for cosine, \(\cos 2A = 2 \cos^2 A - 1\). This identity is also closely related to the half-angle formula for cosine. If we let \(2A = x\), then \(A = x/2\), and the identity becomes \(1 + \cos x = 2 \cos^2 (x/2)\). Taking the square root of both sides gives \(\sqrt{1 + \cos x} = \sqrt{2 \cos^2 (x/2)} = \sqrt{2} |\cos (x/2)|\). This shows how square roots and half angles are connected.

In this problem, we started with an angle \(8\theta\) and successively halved it in each step of simplification (from \(8\theta\) to \(4\theta\), then to \(2\theta\), and finally to \(\theta\)), which is characteristic of applying the half-angle concept derived from the double-angle identity.