Question

If cos A = `5/13`, then verify that `cosA/(1 - tanA) + sinA/(1 - cotA) = cosA + sinA`

Answer 1

Given, cosA = `5/13 = ("Base(B)")/("Hypotenuse(H)")` 

By Pythagoras theorem, we get

H² = P² + B²

∴ (13)² = P² + (5)²

∴ 169 = P² + 25

∴ 169 − 25 = p²

∴ 144 = p²

∴ p = 12

Now, we have to verify that

`cosA/(1 - tanA) + sinA/(1 - cotA) = cos A + sinA`

LHS = `cosA/(1 - tanA) + sinA/(1 - cotA)`

`(5/13)/(1 - 12/5) + (12/13)/(1 - 5/12)`

= `(5/13)/((5 - 12)/5) + (12/13)/((12 - 5)/12)`

= `5/13 xx (-5/7) + 12/13 xx 12/7`

= `-25/91 + 144/91`

= `119/91`

= `17/13`

RHS = cosA + sinA

= `5/13 + 12/13`

= `17/13`

LHS = RHS, Hence proved.