Question

\(1 - \cos^4 \theta = \)

• A. \(\cos^2 \theta (1 - \cos^2 \theta)\)
• B. \(\sin^2 \theta (1 + \cos^2 \theta)\)
• C. \(\sin^2 \theta (1 - \sin^2 \theta)\)
• D. \(\sin^2 \theta (1 + \sin^2 \theta)\)

Hint:

Whenever you see trigonometric functions with even powers (like 2, 4, 6), be on the lookout for opportunities to use the difference of squares factorization or the Pythagorean identities.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem involves simplifying a trigonometric expression by using algebraic factorization and fundamental trigonometric identities.

Step 2: Key Formula or Approach:
We will use the algebraic identity for the difference of squares: \(a^2 - b^2 = (a - b)(a + b)\).
We will also use the Pythagorean identity: \(\sin^2 \theta + \cos^2 \theta = 1\), which can be rearranged to \(\sin^2 \theta = 1 - \cos^2 \theta\).

Step 3: Detailed Explanation:
The given expression is \(1 - \cos^4 \theta\).
We can write this as \(1^2 - (\cos^2 \theta)^2\).
This is a difference of squares, where \(a=1\) and \(b=\cos^2 \theta\). Factoring it gives:
\[ (1 - \cos^2 \theta)(1 + \cos^2 \theta) \] Now, we use the Pythagorean identity \(\sin^2 \theta = 1 - \cos^2 \theta\) to substitute for the first term:
\[ (\sin^2 \theta)(1 + \cos^2 \theta) \] This matches option (B).

Step 4: Final Answer:
The expression \(1 - \cos^4 \theta\) is equal to \(\sin^2 \theta (1 + \cos^2 \theta)\).