Question

If tanA and tanB are the root of X2- px +q =o and the cotA and cotB are the root of X2 -rx+s=0 . find rs????????

Answer 1

★ TRIGONOMETRIC REDUCTIONS ★

Sum of the roots of first and second equation -

TanA + TanB = p

CotA + CotB = r

Product of roots of first and second equation -

TanA ( TanB ) = q

CotA ( CotB ) = s

Now proceed accordingly provided steps in the attachment ...

Results will lead to ,

rs = p : q²

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Answer 2

Answer: rs = p/q².

Explanation:

Let p(x) = x² - px + q = 0 and q(x) = x² - rx + s = 0.

Now, in p(x),

α + β = - b/a = - p

=> tanA + tanB = p _(A)

& αβ = c/a = q

=> tanA·tanB = q _(B)

In q(x),

α + β = - b/a = r

=> cotA + cotB = r _(1)

& αβ = c/a = s

=> cotA·cotB = s _(2)

Multiplying (1) & (2):-

cotA + cotB + cotA·cotB = rs

=> 1/tanA + 1/tanB + 1/tanAtanB = rs

=> rs = (tanA + tanB)/(tanA·tanB)²

=> rs = p/q² [From A & B.]