Question

Prove that:

\[\cot\theta + \tan\theta = cosec\theta \sec\theta\]

Answer 1

\[\cot\theta + \tan\theta\]

\[ = \frac{\cos\theta}{\sin\theta} + \frac{\sin\theta}{\cos\theta}\]

\[ = \frac{\sin^2 \theta + \cos^2 \theta}{\sin\theta\cos\theta}\]

\[ = \frac{1}{\sin\theta\cos\theta} \left( \sin^2 \theta + \cos^2 \theta = 1 \right)\]

\[ = \frac{1}{\sin\theta} \times \frac{1}{\cos\theta}\]

\[ = \text{ cosec } \theta\sec\theta\]