Question

Simplify: $\dfrac{\cos\theta}{1 - \tan\theta} + \dfrac{\sin\theta}{1 - \cot\theta}$

• A. $\cos\theta - \sin\theta$
• B. $\sin\theta - \cos\theta$
• C. $\cos\theta + \sin\theta$
• D. $(1 - \tan\theta)\sin\theta$

Hint:

Rational expressions simplify faster when converted to $\sin\theta, \cos\theta$ terms.

Answer 1

The Correct Option is C

Convert to sin and cos: \[ \tan\theta = \frac{\sin\theta}{\cos\theta},\quad \cot\theta = \frac{\cos\theta}{\sin\theta} \] So: \[ \frac{\cos\theta}{1 - \frac{\sin\theta}{\cos\theta}} + \frac{\sin\theta}{1 - \frac{\cos\theta}{\sin\theta}} = \frac{\cos^2\theta}{\cos\theta - \sin\theta} + \frac{\sin^2\theta}{\sin\theta - \cos\theta} = \frac{\cos^2\theta - \sin^2\theta}{\cos\theta - \sin\theta} = \frac{(\cos\theta + \sin\theta)(\cos\theta - \sin\theta)}{\cos\theta - \sin\theta} \Rightarrow \cos\theta + \sin\theta \]