Understanding the Function f(x) = sin²x + cos²(2x): A Comprehensive Guide

Mathematics is full of elegant identities and surprising relationships—nowhere is this more evident than in the function:
f(x) = sin²x + cos²(2x).
At first glance, this expression blends basic trigonometric components, but beneath its simplicity lies powerful mathematical insights valuable for students, educators, and enthusiasts alike.

In this SEO-optimized article, we unpack f(x) = sin²x + cos²(2x), exploring its identity, simplification, key properties, graph behavior, and practical applications.

Understanding the Context

What Is f(x) = sin²x + cos²(2x)?

The function f(x) combines two fundamental trigonometric terms:

  • sin²x: the square of the sine of x
  • cos²(2x): the square of the cosine of double angle x
f(x) = \sin^2 x + \cos^2(2x)

Key Insights

Both terms involve powers of sine and cosine, but with different arguments, making direct simplification non-obvious.

Step 1: Simplifying f(x) Using Trigonometric Identities

To better understand f(x), we leverage core trigonometric identities.

Recall these foundational rules:

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Final Thoughts

  1. Pythagorean Identity:
    sin²θ + cos²θ = 1
  2. Double Angle Identity for cosine:
    cos(2x) = cos²x - sin²x = 2cos²x - 1 = 1 - 2sin²x
  3. Power-Reducing Identities:
    sin²θ = (1 - cos(2θ))/2
    cos²θ = (1 + cos(2θ))/2

Apply the Power-Reducing Formula to sin²x and cos²(2x)

Start by rewriting each term using identities:

  • sin²x = (1 - cos(2x))/2
  • cos²(2x) = (1 + cos(4x))/2

Now substitute into f(x):

f(x) = sin²x + cos²(2x)
= (1 - cos(2x))/2 + (1 + cos(4x))/2

Combine the terms:

f(x) = [ (1 - cos(2x)) + (1 + cos(4x)) ] / 2
= (2 - cos(2x) + cos(4x)) / 2
= 1 - (cos(2x))/2 + (cos(4x))/2

Thus,
f(x) = 1 + (cos(4x) - cos(2x))/2