Question

J. JIIUVV LIIULULUI LOTIU
4.Prove that (sinA+CosecA)2+(CosA+SecA)2=7+tan?+Cot?A​

Answer 1

Correct Question :-

To Prove :-

$\sf (sinA+cosecA)^2+(cosA+secA)^2= 7+tan^2A +cot^2A$

Solution :-

$\sf L.H.S = (sinA+cosecA)^2(cosA+secA)^2$

        $= \sf sin^2A+cosec^2A+2sinAcosecA+cos^2A+sec^2A+2cosAsecA \\\\= (sin^2A+cos^2A)+cosec^2A+sec^2A+2sinA cosecA+2cosAsecA$

$\bullet \bf \ sin^2A+cos^2A = 1 \\\\\bullet \ cosecA = \dfrac{1}{sinA} \\\\\bullet \ secA = \dfrac{1}{cosA}$

        $= \sf 1+cosec^2A+sec^2A+2 \times sinA \times \dfrac{1}{sinA} +2 \times cosA \times \dfrac{1}{cosA} \\\\= 1+2+2+cosec^2A+sec^2A\\\\= 5+ cosec^2A+sec^2A$

$\bullet \bf \ cosec^2A=cot^2A+1\\\\\bullet \ sec^2A = tan^2A+1$

         $= \sf 5+ cot^2A+1+tan^2A+1 \\\\= 7+tan^2A+cot^2A\\\\= R.H.S$

Hence proved.