Question

\(\frac{1-\cos 2x}{1+\cos 2x}-\sec^2 x=\)

• A. \(1\)
• B. \(\tan 2x\)
• C. \(\sec 2x\)
• D. \(0\)
• E. \(-1\)

Hint:

Memorize identities involving double angles and \(\sec^2 x=1+\tan^2 x\). They help simplify expressions quickly.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This problem requires simplifying a trigonometric expression using the double angle and Pythagorean identities.
Step 2: Key Formula or Approach:
We will use the following trigonometric identities:
1. \(1 - \cos(2x) = 2\sin^2(x)\)
2. \(1 + \cos(2x) = 2\cos^2(x)\)
3. \(\tan^2(x) = \frac{\sin^2(x)}{\cos^2(x)}\)
4. The Pythagorean identity: \(\sec^2(x) - \tan^2(x) = 1\) or \(\tan^2(x) - \sec^2(x) = -1\).
Step 3: Detailed Explanation:
Let's simplify the first part of the expression, \(\frac{1-\cos(2x)}{1+\cos(2x)}\).
Using the double angle identities:
\[ \frac{1-\cos(2x)}{1+\cos(2x)} = \frac{2\sin^2(x)}{2\cos^2(x)} \] Cancel the 2s:
\[ = \frac{\sin^2(x)}{\cos^2(x)} = \tan^2(x) \] Now, substitute this back into the original expression:
\[ \frac{1-\cos(2x)}{1+\cos(2x)} - \sec^2(x) = \tan^2(x) - \sec^2(x) \] Using the Pythagorean identity \(\sec^2(x) - \tan^2(x) = 1\), we can rearrange it to find the value of our expression:
\[ \tan^2(x) - \sec^2(x) = -(\sec^2(x) - \tan^2(x)) = -1 \] Step 4: Final Answer:
The value of the expression simplifies to -1. Therefore, option (E) is the correct answer.