Question

(cosA/1-tanA) + (sinA/1-cotA) = cosA + sinA​

Answer 1

To Prove ;

$\small\bf\dfrac{cosA}{1-tanA} +\dfrac{sinA}{1-cotA} =cosA+sinA$

Formulae Used ;

$\bf\maltese\:\:\: tanA=\dfrac{sinA}{cosA} \\ \\ \bf\maltese\:\:\:cotA=\dfrac{cosA}{sinA} \\ \\\bf\maltese\:\:\:a^{2} -b^{2} =(a-b) (a+b)$

Proof ;

$\huge\mathcal{LHS} :-$

$\sf\dfrac{cosA}{1-tanA} +\dfrac{sinA}{1-cotA}\\ \\ \sf=\dfrac{cosA}{1-\dfrac{sinA}{cosA} } +\dfrac{sinA}{1-\dfrac{cosA}{sinA} }\\ \\\sf=\dfrac{cos^{2}A }{cosA-sinA}+\dfrac{sin^2A}{sinA-cosA} \\ \\ \sf=\dfrac{sin^2A-cos^2A}{sinA-cosA} \\ \\ \sf=\dfrac{(\cancel{sinA-cosA})(sinA+cosA)}{\cancel{sinA-cosA}} \\ \\\bf\Large=sinA+cosA\\ \\ \huge\mathcal{=RHS}$

Hence Proved !