Question

Please answer this question ​

Answer 1

Solution

$l.h.s.= 1 + \frac{cot {}^{2} a}{1 + cosec \: a} \\ = 1 + \frac{ \frac{cos {}^{2}a }{sin {}^{2} a} }{1 + \frac{1}{sin \: a} } \\ = 1 + \frac{ \frac{cos {}^{2} }{sin { }^{2}a } }{ \frac{1 + sin \: a}{sin \: a} } \\ = 1 + \frac{cos {}^{2}a }{sin \: a \: (1 + sin \: a)} \\ = 1 + \frac{cos {}^{2}a }{sin \: a \: + sin {}^{2}a } \\ = \frac{sin \: a + sin {}^{2}a + cos {}^{2}a }{sin \: a \: (1 + sin \: a)} \\ = \frac{(sin \: a \: + 1)}{sin \: a \: (1 + sin \: a)} \\ = \frac{1}{sin \: a} \\ = cosec \: a \: = \: r.h.s.(proved)$