Question

(cosec a - sin a) (seca- cos a)(tana + cot a) = 1
prove using identities​

Answer 1

To Prove :-

$\sf (cosecA-sinA)(secA-cosA)(tanA+cotA) = 1$

Solution :-

$\sf L.H.S = (cosecA-sinA)(secA-cosA)(tanA+cotA)$

$\bullet \bf \ cosecA = \dfrac{1}{sinA} \\\\\bullet \ secA = \dfrac{1}{cosA} \\\\\bullet \ cotA = \dfrac{1}{tanA}$

        $= \sf \left( \dfrac{1}{sinA}-sinA} \right) \left( \dfrac{1}{cosA} -cosA \right) \left( tanA + \dfrac{1}{tanA} \right) \\\\= \left( \dfrac{1-sin^2A} {sinA} \right) \left( \dfrac{1-cos^2A}{cosA} \right) \left( \dfrac{tan^2A+1}{tanA} \right)$

$\bullet \bf \ 1-sin^2A = cos^2A\\\\\bullet \ 1-cos^2A = sin^2A \\\\\bullet \ tan^2A+1 = sec^2A$

          $= \sf \dfrac{cos^2A}{sinA} \times \dfrac{sin^2A}{cosA}\times \dfrac{sec^2A}{tanA} \\\\= cosAsinA \times \dfrac{sec^2A}{tanA}$

$\bullet \bf \ sec^2A = \dfrac{1}{cosA} \\\\\bullet \ tanA = \dfrac{sinA}{cosA}$

           $= \sf cosA sinA \times \dfrac{1}{cos^2A} \times \dfrac{cosA}{sinA} \\\\= \dfrac{cosAsinA}{cosAsinA} \\\\= 1 \\\\= R.H.S$

Hence proved.