Question

In a triangle ABC with ∠C = 90° the equation whose roots are tan A and tan B is ______.

Answer 1

In a triangle ABC with ∠C = 90° the equation whose roots are tan A and tan B is `underline(x^2 - (2/(sin 2A)) x + 1` = 0.

Explanation:

Given a ΔABC with ∠C = 90°

So, the equation whose roots are tanA and tanB is

x² – (tanA + tanB)x + tanA.tanB = 0

A + B = 90°   ......[∵ ∠C = 90°]

⇒ tan(A + B) = tan90°

⇒ `(tanA + tanB)/(1 - tanA tanB) = 1/0`

⇒ 1 – tanA tanB = 0

⇒ tan A tan B = 1   .......(i)

Now tanA + tanB = `sinA/cosA + sinB/cosB`

= `(sinA cosB + cosA sinB)/(cosA cosB)`

= `(sin(A + B))/(cosA cosB)`

= `(sin 90^circ)/(cosA. cos(90^circ - A))`

= `1/(cosA sinA)`

∴ tanA + tanB = `2/(2sinA cosA)`

= `2/(sin 2A)`   ......(ii)

From (i) and (ii) we get

`x^2 - (2/(sin 2A)) x + 1` = 0