Question

The expression \(\cos^2 \theta + \cos^2 (\theta + \phi) - 2 \cos \theta \cos (\theta + \phi)\) is:

• A. independent of \(\theta\)
• B. independent of \(\phi\)
• C. independent of \(\theta\) and \(\phi\)
• D. dependent on \(\theta\) and \(\phi\)

Hint:

When dealing with trigonometric expressions involving sums or differences of angles, try using known identities to simplify the expression.

Answer 1

The Correct Option is B

Step 1: We are given the expression:

\[ \cos^2 \theta + \cos^2(\theta + \phi) - 2 \cos \theta \cos(\theta + \phi). \]

Step 2: Use trigonometric identities to simplify the expression. First, expand \( \cos(\theta + \phi) \) using the angle addition formula:

\[ \cos(\theta + \phi) = \cos \theta \cos \phi - \sin \theta \sin \phi. \]

Step 3: Substitute this into the given expression and simplify. After simplification, you’ll see that the expression is independent of \( \phi \).

Step 4: Therefore, the expression is independent of \( \phi \), making the correct answer B.