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If A B C ,sinC+cosC+sin(2B+C)-cos(2B+C)...

If ` A B C ,sinC+cosC+sin(2B+C)-cos(2B+C))=2sqrt(2.)` Prove that ` A B C` is right-angled isosceles.

Text Solution

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`sinC+cos C+sin(2B+C)-cos(2B+C)=2sqrt(2)`
`rArr sinC+sin(2B+C)+cosC-cos(2B+C)=2sqrt(2)`
`rArr 2sin(B+C)cosB+2 sin(B+C)sinB=2sqrt(2)`
`rArr 2sin A (cos B+sinB)=2sqrt(2) [because sin(B+C)=sinA]`
`rArr 2sqrt(2)sin A cos((pi)/(4)-B)=2sqrt(2)`
`rArr sin A cos (B-(pi)/(4))=1`
`rArr sinA=1 "and" cos(B-(pi)/(4))=1`
`rArr A=(pi)/(2) "and" B=(pi)/(4)`.
Hence, `C=(pi)/(4)`
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