First slide

Trigonometric Identities

Question

If sec $α$ and cosec $α$ are the roots of the equation $x^{2} - a x + b = 0,$ then

Moderate

A

$a^{2} = b (b - 2)$
B

$a^{2} = b (b + 2)$
C

$a^{2} + b^{2} = 2 b$
D

none of these

Solution

It is given that sec $α$ and cosec $α$ are the roots of the equation $x^{2} - a x + b = 0$
$∴ \sec α + cosec α = a and \sec α cosec α = b \Rightarrow \sin α + \cos α = a \sin α \cos α and \sin α \cos α = \frac{1}{b} \Rightarrow \sin α + \cos α = \frac{a}{b} and \sin α \cos α = \frac{1}{b} (\sin α + \cos α)^{2} = 1 + 2 \sin α \cos α \Rightarrow \frac{a^{2}}{b^{2}} = 1 + \frac{2}{3} \Rightarrow a^{2} = b (b + 2)$

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