Question

Simplify the expression: \[ \frac{\sin \theta(1 + \tan \theta) + \cos \theta (1 + \cot \theta)}{\csc \theta - \sin \theta} \cdot \frac{\sec \theta}{\cos \theta (\tan \theta + \cot \theta)} = ? \]

• A. \( \sin \theta \cos \theta \)
• B. \( \csc \theta \sec \theta \)
• C. \( \csc \theta + \sec \theta \)
• D. \( \sin \theta + \cos \theta \)

Hint:

When simplifying complex trigonometric expressions, break down the terms using standard identities and look for ways to combine similar terms.

Answer 1

The Correct Option is C

We are given the expression: \[ \frac{\sin \theta(1 + \tan \theta) + \cos \theta(1 + \cot \theta)}{(\csc \theta - \sin \theta)(\sec \theta)(\cos \theta (\tan \theta + \cot \theta))} \] We begin by simplifying each term in the numerator and the denominator. - Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). - Substituting these into the expression: \[ \sin \theta (1 + \frac{\sin \theta}{\cos \theta}) + \cos \theta (1 + \frac{\cos \theta}{\sin \theta}) \] This simplifies as follows: \[ = \sin \theta \left(\frac{\cos \theta + \sin \theta}{\cos \theta}\right) + \cos \theta \left(\frac{\sin \theta + \cos \theta}{\sin \theta}\right) \] \[ = \frac{\sin \theta (\cos \theta + \sin \theta)}{\cos \theta} + \frac{\cos \theta (\sin \theta + \cos \theta)}{\sin \theta} \] Thus, the numerator simplifies to: \[ \frac{\sin \theta (\cos \theta + \sin \theta)}{\cos \theta} + \frac{\cos \theta (\sin \theta + \cos \theta)}{\sin \theta} \] - The denominator involves the terms \( \csc \theta - \sin \theta \), \( \sec \theta \), and \( \tan \theta + \cot \theta \). - Using the identities \( \sec \theta = \frac{1}{\cos \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \), the expression becomes: \[ (\frac{1}{\sin \theta} - \sin \theta) \cdot \frac{1}{\cos \theta} \cdot \cos \theta \cdot \left( \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} \right) \] This simplifies to: \[ \frac{1 - \sin^2 \theta}{\sin \theta} \cdot \frac{1}{\cos \theta} \cdot \left(\frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta}\right) \] - We now use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to further simplify the terms, and the expression simplifies to \( \csc \theta + \sec \theta \). Thus, the value of the expression is: \[ \boxed{\csc \theta + \sec \theta} \]