Question

evaluate (1+tan theta + sec theta)(1+cot theta - cosec theta)​

Answer 1

$\underline\bold{ \large { \: \: \: Answer \: \: \: }}$

$\implies$ 2

$\underline\bold{ \large { \: \: \:Explanation \: \: \: }}$

$\to(1 + tan \: \theta + sec \: \theta) \: (1 + cot \: \theta \: - cosec \: \theta) \\ \\ \to(1 + \frac{sin \: \theta}{cos \: \theta} + \frac{1}{cos \: \theta} ) \: (1 + \frac{cos \: \theta}{ sin \: \theta} - \frac{1}{sin \: \theta} ) \\ \\ \to( \frac{cos \: \theta + sin \: \theta + 1}{cos \: \theta}) \: ( \frac{sin \: \theta + cos \: \theta - 1}{sin \: \theta} ) \\ \\ \to \frac{sin \: \theta \times cos \: \theta + {cos}^{2} \: \theta - cos \: \theta + {sin}^{2} \: \theta + sin \: \theta \times cos \: \theta - sin \: \theta + sin \: \theta + cos \: \theta - 1 }{sin \: \theta \: \times cos \: \theta} \\ \\ \to \frac{(sin \: \theta \times cos \: \theta + sin \: \theta \times cos \: \theta )+( {cos}^{2} \: \theta \: + {sin}^{2} \: \theta \: )}{sin \: \theta \: \times cos \: \theta \: } \\ \\ \to \frac{2 \times sin \: \theta \times cos \: \theta + 1 - 1}{sin \: \theta \: \times cos \: \theta} \\ \\ \to 2$