Question

Prove that √1+cosA/√1-cosA=cosecA+cotA.​

Answer 1

TO PROVE :-

$\\ \sf \: \sqrt{ \dfrac{1 + cosA}{1 - cosA} } = cosecA + cotA$

SOLUTION :-

$\\ \sf \: R.H.S = \sqrt{ \dfrac{1 + cosA}{1 - cosA} }$

Multiplying numerator and denominator with √1+cosA ,

$\\ \implies\sf \: \sqrt{ \dfrac{1 + cosA}{1 - cosA} \times \dfrac{1 + cosA}{1 + cosA} } \\ \\ \\ \implies\sf \: \sqrt{ \dfrac{(1 + {cosA)}^{2} }{(1 - cosA)(1 + cosA)} } \\ \\ \\ \implies \sf \: \sqrt{ \dfrac{(1 + {cosA)}^{2} }{1 - {cos}^{2}A } }$

$\\ \\ \boxed{ \bf \: {sin}^{2}A + {cos}^{2}A = 1 } \\ \\ \\ \implies\sf \: \sqrt{ \dfrac{(1 + {cosA)}^{2} }{ {sin}^{2}A } } \\ \\ \\ \implies \sf \: \dfrac{1 + cos}{sin} \\ \\ \\ \implies \sf \: \dfrac{1}{sinA} + \dfrac{cosA}{sinA}$

$\\ \boxed{ \bf \: \dfrac{1}{sinA} = cosecA} \: \: \: \: \: \: \: \: \boxed{ \bf \: \dfrac{cosA}{sinA} = cotA } \\ \\ \\ \implies \sf \: cosecA + cotA \: = R.H.S \: \: \: \: (verified)$

MORE TO KNOW :-

★ 1 + tan²A = sec²A

★ 1 + cot²A = cosec²A

★ tanA = sinA/cosA

★ cosA = 1/secA