First slide

Trigonometric Identities

Question

A quadratic equation whose roots are $\cos e c^{2} θ$ and $\sec^{2} θ$ can be

Moderate

A

$x^{2} - 2 x + 2 = 0$
B

$x^{2} - 3 x + 3 = 0$
C

$x^{2} - x + 1 = 0$
D

None of these

Solution

$\begin{array}{l} S u m o f r o o t s = \sec^{2} θ + \cos e c^{2} θ = \frac{1}{\cos^{2} θ} + \frac{1}{\sin^{2} θ} \\ = \frac{4}{\sin^{2} 2 θ} \geq 4 \\ A l s o, p r o d u c t o f r o o t s = \sec^{2} θ \cos e c^{2} θ = \frac{4}{\sin^{2} 2 θ} \geq 4 \end{array}$
$∴$ The required quadratic equation will be $x^{2} - t x + t = 0$ where $t \geq 4$ .

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