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Question

In a ABC, if cosA+cosB+cosC=32, then the triangle is

A
an isosceles
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B
an equilateral
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C
a scalene
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D
a right angled
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Solution

The correct option is B an equilateral
In a triangle
A+B+C=π
cosA+cosB+cosC=2cosA+B2cosAB2+cosC=2cos(π2C2)cosAB2+12sin2C2=2sin2C2+2sinC2cosAB2+1
2sin2C2+2sinC2cosAB2+1=324sin2C24sinC2cosAB2+1=0(1)

Now, we know that
sinC2R
Δ016cos2AB2160cos2AB210cos2AB21
But cos2θ1, so
cos2AB2=1A=B
Using equation (1),
4sin2C24sinC2+1=0(2sinC21)2=0sinC2=12C=π3
A=B=C=π3

Alternate solution:
By observation,
cosA=cosB=cosC=12
Satisfies the given equation
A=B=C=60
Therefore, the triangle is equilateral triangle.

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