Question

\(\cos^2 A (1 + \tan^2 A) = \)

• A. \(\sin^2 A\)
• B. \(\csc^2 A\)
• C. 1
• D. \(\tan^2 A\)

Hint:

Memorizing the three Pythagorean identities (\(\sin^2\theta + \cos^2\theta = 1\), \(1 + \tan^2\theta = \sec^2\theta\), and \(1 + \cot^2\theta = \csc^2\theta\)) is crucial for quickly solving trigonometric simplification problems.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem requires simplifying a trigonometric expression using fundamental Pythagorean identities.

Step 2: Key Formula or Approach:
We will use the Pythagorean identity:
\[ 1 + \tan^2 A = \sec^2 A \] And the reciprocal identity:
\[ \sec A = \frac{1}{\cos A} \]

Step 3: Detailed Explanation:
We start with the given expression:
\[ \cos^2 A (1 + \tan^2 A) \] First, substitute the Pythagorean identity \(1 + \tan^2 A = \sec^2 A\) into the expression:
\[ = \cos^2 A (\sec^2 A) \] Next, use the reciprocal identity \(\sec A = \frac{1}{\cos A}\), which means \(\sec^2 A = \frac{1}{\cos^2 A}\):
\[ = \cos^2 A \left( \frac{1}{\cos^2 A} \right) \] Now, cancel out the \(\cos^2 A\) terms:
\[ = 1 \]

Step 4: Final Answer:
The value of the expression \(\cos^2 A (1 + \tan^2 A)\) is 1.