If sin- sin- sin- are in A-P and cos- cos- cos- are in G-P then cos2-cos2-4cos-cos-1-sin-sin-

From the given conditions we have 2 sinβ = sinα + sinγ→ (i), cos^2 β = cosαcosγ→ (ii) Squaring (i), 4 sin^2 β = sin^2 α + sin^2 γ + 2 sinαsinγ Using (ii), 4(1 c ...

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Standard XII

Mathematics

Basic Trigonometric Identities

If sinα, sinβ...

Question

If $sin α, sin β, sin γ$ are in A.P and $cos α, cos β, cos γ$ are in G.P then $\frac{{cos}^{2} α + {cos}^{2} γ - 4 cos α cos γ}{1 - sin α sin γ}$

-2

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Solution

The correct option is A -2
From the given conditions we have 2 $sin β = sin α + sin γ \to (i), {cos}^{2} β = cos α cos γ \to (i i)$
Squaring (i), $4 {sin}^{2} β = {sin}^{2} α + {sin}^{2} γ + 2 sin α sin γ$
Using (ii), $4 (1 - cos α cos γ) = 1 - {cos}^{2} α + 1 - {cos}^{2} γ + 2 sin α sin γ$
$\Rightarrow {cos}^{2} α + {cos}^{2} γ - 4 cos α cos γ = 2 (sin α sin γ - 1) \Rightarrow \frac{{cos}^{2} α + {cos}^{2} γ - 4 cos α cos γ}{1 - sin α sin γ} = - 2$

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Q. If

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\frac{{cos}^{2} α + {cos}^{2} γ - 4 cos α cos γ}{1 - sin α sin γ}

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If sinα,sinβ,sinγ are in A.P and cosα,cosβ,cosγ are in G.P then cos2α+cos2γ−4cosαcosγ1−sinαsinγ

If $sin α, sin β, sin γ$ are in A.P and $cos α, cos β, cos γ$ are in G.P then $\frac{{cos}^{2} α + {cos}^{2} γ - 4 cos α cos γ}{1 - sin α sin γ}$