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