Question

sinA (1+tanA)+cosA (1+cotA)=secA+cosecA​

Answer 1

To Prove :-

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

Solution :-

$\sf L.H.S = sinA(1+tanA)+cosA(1+cotA)$

        $= \sf sinA+SinAtanA+cosA+cosAcotA$

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

       $= \sf sinA+ sinA \times \dfrac{sinA}{cosA} +cosA + cosA \times \dfrac{cosA}{sinA} \\\\= sinA +\dfrac{sinA^2}{cosA} +cosA+\dfrac{cos^2A}{sinA} \\\\= \left( sinA + \dfrac{cos^2A}{sinA} \right) + \left( cosA+\dfrac{sin^2A}{cosA} \right) \\\\= \left( \dfrac{sin^2A+cos^2A}{sinA} \right) +\left( \dfrac{cos^2A+sin^2A}{cosA} \right)$

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

       $= \sf \dfrac{1}{sinA} +\dfrac{1}{cosA} \\\\= cosecA+secA \\\\= secA+cosecA \\\\= R.H.S$

Hence proved.