Question

Prove that root sec theta-1 / root sec theta + root sec theta +1 / sec theta -1=2 cosec theta

Answer 1

We have :

LHS:

$\bold{ \huge{ \sqrt{ \frac{ \sec(a) - 1 }{ \sec(a) + 1 } }} + \sqrt{ \frac{ \sec(a) + 1 }{ \sec(a ) - 1 } }}$

$\bold{ \huge{ \frac{ \sqrt{ \sec(a) - 1} }{ \sqrt{ \sec(a + 1 } }} + \bold{ \frac{ \sqrt{ \sec(a) + 1} }{ \sqrt{ \sec(a) - 1 } }}}$

$\bold{ \huge{ \frac{(sec \: a - 1) + (sec \: a + 1)}{ \sqrt{(sec \: a + 1)(sec \: a - 1)} }}}$

$\bold{ \huge{ \frac{2sec \: a}{ \sqrt{ {sec }^{2}a - 1 } }}}$

$\bold{ \huge{ \ \frac{2sec \: a}{ \sqrt{ {tan}^{2} a} }}}$

$\bold{ \huge{ 2 \: sec \: a \times cot \: a}}$

$\bold{ \huge( \frac{2}{cos \: a}} {\bold{ \huge\times \frac{cos \: a}{sin \: a} )}}$

$\bold{ \huge{ \frac{2}{sin \: a}}}$



$\bold{ \huge{2 \: cosec \: a}}$

$\bold{ \huge{ \color{red}{proved:-}}}$