Question

if cosec A+cot A=5 then evaluate sec A​

Answer 1

Answer:

$\cosec(a) + \cot(a) = 5 .......(1)\\ since \\ { \cosec }^{2} a - {cot}^{2} a = 1 \\ (cosec \: a + cot \: a)(cosec \: a - cot \: a) = 1 \\ 5 \times (cosec \: a - cot \: a) = 1 \\ cosec \: a - cot \: a = \frac{1}{5} .......(2) \\ adding \: 1 \: and \: 2 \\ 2 \: cosec \: a = 5 + \frac{1}{5} = \frac{26}{5} \\ cosec \: a = \frac{26}{5 \times 2} = \frac{13}{5} \\ sin \: a = \frac{5}{13} \\ cos \: a = \sqrt{1 - {sin}^{2} a} \\ = \sqrt{1 - ({ \frac{5}{13} })^{2} } \\ = \sqrt{ \frac{169 - 25}{169} } \\ = \sqrt{ \frac{144}{169} } = \frac{12}{13} \\ so \\ sec \: a = \frac{13}{12}$