Question

Prove that : cos A / (1-tanA) + sinA /( 1-cotA) = sinA + cos A ​

Answer 1

$\sf\large\underline\blue{Given:-}$

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

$\sf\large\underline\blue{To\: Prove :-}$

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

$\sf\large\underline\blue{Formula \: Used :-}$

• a² - b² = ( a+b) ( a -b)

$\sf\large\underline\blue{Solution :-}$

L. H. S

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

=$\dfrac{cosA}{1-\dfrac{sinA}{cosA}}+\dfrac{sinA}{\dfrac{cosA}{sinA}}$

=$\dfrac{cosA}{\dfrac{cosA-sinA}{cosA}}+\dfrac{sinA}{\dfrac{sinA-cosA}{sinA}}$

=$\dfrac{cos^{2}A}{cosA-sinA}+\dfrac{sin^{2}A}{sinA-cosA}$

=$\dfrac{(-1) cos^{2}A}{(-1) (cosA-sinA) }+\dfrac{sin^{2}A}{sinA-cosA}$

=$\dfrac{sin^{2}A-cos^{2}A}{cosA-sinA}$

=$\dfrac{(sinA+cosA) (\cancel{sinA-cosA}) }{\cancel{sinA-cosA}}$

=$sinA+cosA$

= R. H. S

L.H.S = R.H.S

Hence Proved.