Question

If tan alpha and tan beta are the roots of x^2-px+q=0, cot alpha and cot beta are the roots of x^2-rx+s=0 , then rs is necessarily A) pq B)1/pq C) p/q^2 D) q/p^2

Answer 1

if tanA and tanB are roots of x^2 - px + q = 0
then , tanA + tanB = -(-p) = p .....- (i)
 also tanA * tanB = q .... - (ii)
now , cotA and cotB are zeroes of x^2 -rx + s = 0
therefore, cotA * cotB = s
therefore 1 / (tanA * tanB) = s
therefore 1/q = s (from (ii)) ....... -(iii)
now, cotA + cotB = -(-r)
 1/tanA + 1/tanB = r
(tanB + tanA)/(tanA * tanB) = r 
p / q = r (from (i) and (ii)) .......- (iv)
therefore r * s = p/q * 1/q (from (iii) and (iv))
r * s = p/q^2
therefore option C is the right answer