Question

The value of \( (\text{cosec}^2\theta - \cot^2\theta)(1 - \cos^2\theta) \) is

• A. \( \sec^2\theta \)
• B. \( \tan^2\theta \)
• C. \( \text{cosec}^2\theta \)
• D. \( \sin^2\theta \)

Hint:

Always look for fundamental identities when you see squared trigonometric functions. The three Pythagorean identities (\(\sin^2\theta + \cos^2\theta = 1\), \(1 + \tan^2\theta = \sec^2\theta\), and \(1 + \cot^2\theta = \text{cosec}^2\theta\)) are the keys to simplifying most such expressions.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem requires the application of fundamental trigonometric identities, specifically the Pythagorean identities.
Step 2: Key Formula or Approach:
We will use the following two Pythagorean identities:
1. \( \sin^2\theta + \cos^2\theta = 1 \), which can be rearranged to \( 1 - \cos^2\theta = \sin^2\theta \).
2. \( 1 + \cot^2\theta = \text{cosec}^2\theta \), which can be rearranged to \( \text{cosec}^2\theta - \cot^2\theta = 1 \).
Step 3: Detailed Explanation:
The given expression is \( (\text{cosec}^2\theta - \cot^2\theta)(1 - \cos^2\theta) \).
Let's simplify each part of the expression separately.
Part 1: \( (\text{cosec}^2\theta - \cot^2\theta) \)
From the Pythagorean identity \( 1 + \cot^2\theta = \text{cosec}^2\theta \), we get:
\[ \text{cosec}^2\theta - \cot^2\theta = 1 \] Part 2: \( (1 - \cos^2\theta) \)
From the Pythagorean identity \( \sin^2\theta + \cos^2\theta = 1 \), we get:
\[ 1 - \cos^2\theta = \sin^2\theta \] Now, substitute these simplified parts back into the original expression:
\[ (\text{cosec}^2\theta - \cot^2\theta)(1 - \cos^2\theta) = (1)(\sin^2\theta) = \sin^2\theta \] Step 4: Final Answer:
The value of the expression is \( \sin^2\theta \).