Prove that root sec theta-1 / root sec theta + root sec theta +1 / sec theta -1=2 cosec theta
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We have :
LHS:
![\bold{ \huge{ \sqrt{ \frac{ \sec(a) - 1 }{ \sec(a) + 1 } }} + \sqrt{ \frac{ \sec(a) + 1 }{ \sec(a ) - 1 } }} \bold{ \huge{ \sqrt{ \frac{ \sec(a) - 1 }{ \sec(a) + 1 } }} + \sqrt{ \frac{ \sec(a) + 1 }{ \sec(a ) - 1 } }}](https://tex.z-dn.net/?f=+%5Cbold%7B+%5Chuge%7B+%5Csqrt%7B+%5Cfrac%7B+%5Csec%28a%29+-+1+%7D%7B+%5Csec%28a%29+%2B+1+%7D+%7D%7D+%2B+%5Csqrt%7B+%5Cfrac%7B+%5Csec%28a%29+%2B+1+%7D%7B+%5Csec%28a+%29+-+1+%7D+%7D%7D)
![\bold{ \huge{ \frac{ \sqrt{ \sec(a) - 1} }{ \sqrt{ \sec(a + 1 } }} + \bold{ \frac{ \sqrt{ \sec(a) + 1} }{ \sqrt{ \sec(a) - 1 } }}} \bold{ \huge{ \frac{ \sqrt{ \sec(a) - 1} }{ \sqrt{ \sec(a + 1 } }} + \bold{ \frac{ \sqrt{ \sec(a) + 1} }{ \sqrt{ \sec(a) - 1 } }}}](https://tex.z-dn.net/?f=+%5Cbold%7B+%5Chuge%7B+%5Cfrac%7B+%5Csqrt%7B+%5Csec%28a%29+-+1%7D+%7D%7B+%5Csqrt%7B+%5Csec%28a+%2B+1+%7D+%7D%7D+%2B+%5Cbold%7B+%5Cfrac%7B+%5Csqrt%7B+%5Csec%28a%29+%2B+1%7D+%7D%7B+%5Csqrt%7B+%5Csec%28a%29+-+1+%7D+%7D%7D%7D)
![\bold{ \huge{ \frac{(sec \: a - 1) + (sec \: a + 1)}{ \sqrt{(sec \: a + 1)(sec \: a - 1)} }}} \bold{ \huge{ \frac{(sec \: a - 1) + (sec \: a + 1)}{ \sqrt{(sec \: a + 1)(sec \: a - 1)} }}}](https://tex.z-dn.net/?f=+%5Cbold%7B+%5Chuge%7B+%5Cfrac%7B%28sec+%5C%3A+a+-+1%29+%2B+%28sec+%5C%3A+a+%2B+1%29%7D%7B+%5Csqrt%7B%28sec+%5C%3A+a+%2B+1%29%28sec+%5C%3A+a+-+1%29%7D+%7D%7D%7D)
![\bold{ \huge{ \frac{2sec \: a}{ \sqrt{ {sec }^{2}a - 1 } }}} \bold{ \huge{ \frac{2sec \: a}{ \sqrt{ {sec }^{2}a - 1 } }}}](https://tex.z-dn.net/?f=+%5Cbold%7B+%5Chuge%7B+%5Cfrac%7B2sec+%5C%3A+a%7D%7B+%5Csqrt%7B+%7Bsec+%7D%5E%7B2%7Da+-+1+%7D+%7D%7D%7D+)
![\bold{ \huge{ \ \frac{2sec \: a}{ \sqrt{ {tan}^{2} a} }}} \bold{ \huge{ \ \frac{2sec \: a}{ \sqrt{ {tan}^{2} a} }}}](https://tex.z-dn.net/?f=+%5Cbold%7B+%5Chuge%7B+%5C+%5Cfrac%7B2sec+%5C%3A+a%7D%7B+%5Csqrt%7B+%7Btan%7D%5E%7B2%7D+a%7D+%7D%7D%7D)
![\bold{ \huge{ 2 \: sec \: a \times cot \: a}} \bold{ \huge{ 2 \: sec \: a \times cot \: a}}](https://tex.z-dn.net/?f=+%5Cbold%7B+%5Chuge%7B+2+%5C%3A+sec+%5C%3A+a+%5Ctimes+cot+%5C%3A+a%7D%7D)
![\bold{ \huge( \frac{2}{cos \: a}} {\bold{ \huge\times \frac{cos \: a}{sin \: a} )}} \bold{ \huge( \frac{2}{cos \: a}} {\bold{ \huge\times \frac{cos \: a}{sin \: a} )}}](https://tex.z-dn.net/?f=+%5Cbold%7B+%5Chuge%28+%5Cfrac%7B2%7D%7Bcos+%5C%3A+a%7D%7D+%7B%5Cbold%7B+%5Chuge%5Ctimes+%5Cfrac%7Bcos+%5C%3A+a%7D%7Bsin+%5C%3A+a%7D+%29%7D%7D)
![\bold{ \huge{ \frac{2}{sin \: a}}} \bold{ \huge{ \frac{2}{sin \: a}}}](https://tex.z-dn.net/?f=+%5Cbold%7B+%5Chuge%7B+%5Cfrac%7B2%7D%7Bsin+%5C%3A+a%7D%7D%7D)
![\bold{ \huge{2 \: cosec \: a}} \bold{ \huge{2 \: cosec \: a}}](https://tex.z-dn.net/?f=+%5Cbold%7B+%5Chuge%7B2+%5C%3A+cosec+%5C%3A+a%7D%7D)
LHS:
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