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Statement-1: In a triangle ABC, if sin^(...

Statement-1: In a triangle ABC, if `sin^(2)A + sin^(2)B + sin^(2)C = 2`, then one of the angles must be 90 °.
Statement-2: In any triangle ABC
cos 2A + cos 2B + cos 2C = -1 - 4 cos A cos B cos C

A

Statement-1 is True, Statement-2 is true, Statement-2 is a correct explanation for Statement-1.

B

Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.

C

Statement-1 is True, Statement-2 is False.

D

Statement-1 is False, Statement- 2 is True.

Text Solution

Verified by Experts

In a triangle ABC, we have A+ B +C = `pi`
`thereforecos 2A + cos 2B + cos 2C`
`= 2 cos (A + B} cos (A - B} + 2 cos^(2) C - 1`
`= - 2 cos C cos (A - B) + 2 cos^(2) C - 1`
`= - 2 cos C {cos (A - B) - cos C} - 1`
`= - 2 cos C {cos (A -B) + cos (A+ B)}-1`
`= - 2 cos C {2 cos A cos B} -1`
= -1 - 4 cos A cos B cos C
So, statement-2 is true.
And,
`sin^(2) A + sin^(2) B + sin^(2) C = 2`
`rArr` 1 - cos2A+1 -cos2B+1 - cos2C = 4
`rArr` cos 2A + cos 2B + cos 2C = -1
`rArr` 4cosA cos BcosC = 0 [Using statement- 2]
`rArr` cos A = 0 or cos B = 0 or cos C = 0
`rArrA=pi/2` or `B=(pi)/2`
So, statement-I is true and statement-2 is a correct explanation for statement-I.
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Knowledge Check

  • Statement I If in a triangle ABC sin ^(2) A+sin ^(2)B+sin ^(2)C=2, then one of the angle must be 90^(@). Statement II In any triangles ABC cos 2A+ cos 2B+cos 2C=-1-4 cos A cos B cos C

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    B
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    Statement I is correct but Statement II is incorrect
    D
    Statement I is correct but Statement I is incorrect

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