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The given equation is:

\[
\cos^6{x} + \sin^6{x} - \cos^2{2x} = \frac{1}{16}
\]

Let's try to simplify and solve this step-by-step.

### Step 1: Use trigonometric identities
We will begin by using known trigonometric identities to simplify the terms:

- \(\cos^6{x} + \sin^6{x}\) can be rewritten using the identity:
  \[
  \cos^6{x} + \sin^6{x} = (\cos^2{x} + \sin^2{x})^3 - 3\cos^2{x}\sin^2{x}(\cos^2{x} + \sin^2{x})
  \]
  Since \(\cos^2{x} + \sin^2{x} = 1\), this simplifies to:
  \[
  \cos^6{x} + \sin^6{x} = 1 - 3\cos^2{x}\sin^2{x}
  \]

- Also, for \(\cos^2{2x}\), use the double angle identity:
  \[
  \cos{2x} = \cos^2{x} - \sin^2{x}
  \]
  So,
  \[
  \cos^2{2x} = (\cos^2{x} - \sin^2{x})^2
  \]

### Step 2: Substitute and simplify
Substitute these identities into the original equation:
\[
1 - 3\cos^2{x}\sin^2{x} - (\cos^2{x} - \sin^2{x

This page provides a detailed solution to the trigonometric equation cos^6 x + sin^6 x - cos^2 2x = 1/16. The solution uses key trigonometric identities such as the Pythagorean Identity and Double Angle Identity to simplify the expression. Ideal for undergraduate students studying trigonometry.

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