First slide

Trigonometric Identities

Question

If $x \cos α + y sinα = x \cos β + y sinβ = 2 a$ , then $cosα cosβ$ is

Moderate

A

$\frac{4 a}{x^{2} + y^{2}}$
B

$\frac{4 a^{2} - y^{2}}{x^{2} + y^{2}}$
C

$\frac{4 ay}{x^{2} + y^{2}}$
D

$\frac{4 a^{2} - x^{2}}{x^{2} + y^{2}}$

Solution

we have, $x \cos α + y sinα = x \cos β + y sinβ = 2 a$
$\Rightarrow α, β$ are roots of $x \cos θ + y sinθ = 2 a$
now,
$x \cos θ + y sinθ = 2 a$
${(x \cos θ - 2 a)}^{2} = y^{2} \sin^{2} θ$
$x^{2} \cos^{2} θ - 4 ax cosθ + 4 a^{2} = y^{2} (1 - \cos^{2} θ)$
$\Rightarrow (x^{2} + y^{2}) \cos^{2} θ - 4 ax cosθ + 4 a^{2} - y^{2} = 0$
Clearly, $cosα, cosβ$ are the roots of this equation
$cosα cosβ =$ $\frac{4 a^{2} - y^{2}}{x^{2} + y^{2}}$

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