Question

Evaluate: \[ \frac{\sin \theta (1 + \tan \theta) + \cos \theta (1 + \cot \theta)}{(\cos \theta - \sin \theta)(\sec \theta - \cos \theta)(\tan \theta + \cot \theta)} \]

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

Hint:

Break complex expressions using trigonometric identities. If direct simplification gets messy, try standard angle substitution (e.g., \( \theta = 45^\circ \)).

Answer 1

The Correct Option is C

Start with numerator: \[ \sin \theta (1 + \tan \theta) + \cos \theta (1 + \cot \theta) = \sin \theta + \sin \theta \tan \theta + \cos \theta + \cos \theta \cot \theta \] Use identities: \[ \tan \theta = \frac{\sin \theta}{\cos \theta},\quad \cot \theta = \frac{\cos \theta}{\sin \theta} \Rightarrow \text{numerator} = \sin \theta + \frac{\sin^2 \theta}{\cos \theta} + \cos \theta + \frac{\cos^2 \theta}{\sin \theta} \] Denominator: \[ (\cos \theta - \sin \theta)(\sec \theta - \cos \theta)(\tan \theta + \cot \theta) \] Noticing complexity, test \( \theta = 45^\circ \): Then: \[ \sin \theta = \cos \theta = \frac{1}{\sqrt{2}},\quad \tan \theta = \cot \theta = 1,\quad \sec \theta = \csc \theta = \sqrt{2} \] Numerator: \[ \frac{1}{\sqrt{2}}(1 + 1) + \frac{1}{\sqrt{2}}(1 + 1) = 2 \cdot \frac{1}{\sqrt{2}} \cdot 2 = \frac{4}{\sqrt{2}} = 2\sqrt{2} \] Denominator: \[ % Option (0)(\sqrt{2} - \frac{1}{\sqrt{2}})(1 + 1) = 0 \Rightarrow \text{undefined at } \theta = 45^\circ \] Try algebraic manipulation — numerator becomes: \[ \sin \theta + \cos \theta + \frac{\sin^2 \theta}{\cos \theta} + \frac{\cos^2 \theta}{\sin \theta} = \sin \theta + \cos \theta + \sin \theta \tan \theta + \cos \theta \cot \theta = \csc \theta + \sec \theta \]