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A triangle having vertices as P (cosα, sinα), Q (cosβ, sinβ), R (cos $γ$ , sin $γ$ ), whose orthocenter is (0, 0). Then, the value of cos (α−β) +cos (β− $γ$ ) +cos ( $γ$ −α) is:

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-3/2

-1/2

1/2

3/2

answer is A.

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Detailed Solution

Given, P (cosα, sinα), Q (cosβ, sinβ), R (cos

γ

, sin

γ

) are vertices of In

∆ PQR

.
Here, OP=OQ=OR, where O is orthocenter
∴O (0, 0) is the circumcenter of ΔPQR.
Thus, orthocenter and circumcenter coincide.
Here, Centroid of ΔPQR= (0, 0)

\frac{\cos α + \cos β + \cos γ}{3} = 0

And

\frac{\sin α + \sin β + \sin γ}{3} = 0

Now,

\cos α + \cos β + \cos γ = 0 \dots \dots \dots \dots \dots . (1)

and \sin α + \sin β + \sin γ = 0

………………(2)

\cos α + \cos β = - \cos γ

and si n α + si n β = - \sin γ

………………(2)
by Squaring and adding (1) and (2)_

\cos (α - β) + \cos (β - γ) + \cos (γ - α) = - \frac{3}{2}

Correct option is 1.

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