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In any DeltaABC, prov "If in a "DeltaABC...

In any `DeltaABC`, prov `"If in a "DeltaABC, /_C=90^(@)," then prove that sin"(A-B)+((a^(2)-b^(2)))/((a^(2)+b^(2)))`

A

right angled

B

isosceles

C

either (a) or (b) both

D

neither (a) nor (b)

Text Solution

Verified by Experts

The correct Answer is:
C
Let `(a)/(sinA)=(b)/(sinB)=(c)/(sinC)=k` [say]
Then , a `=ksinA,b=ksinBandc=ksinC` …(i)
We have , `(a^(2)-b^(2))/(a^(2)+b^(2))=(sin(A-B))/(sin(A+B))`
`rArr(k^(2)sin^(2)A-k^(2)sin^(2)B)/(k^(2)sin^(2)A+k^(2)sin^(2)B)=(sin(A-B))/(sin(A+B))` [from Eq . (i)]
`rArr(sin(A-B))/(sin(pi-C))=(sin(A+B)sin(A-B))/(sin^(2)A+sin^(2)B)`
`[because a+B+C=pirArrC=pi-(A+B)andsin^(2)A-sin^(2)B+sin(A+B)sin(A-B)]`
`rArr(sin(A-B))/(sinC)=(sin(pi-C)sin(A-B))/(sin^(2)A+sin^(2)B)`
`rArr(sin(A-B))/(sinC)=(sinCsin(A-B))/(sin^(2)A+sin^(2)B)`
`rArrsin(A-B)[(1)/(sinC)-(sinC)/(sin^(2)A+sin^(2)B)]=0`
`rArr` Either sin (A-B) =0
or `(1)/(sinC)-(sinC)/(sin^(2)A+sin^(2)B)=0`
`rArr"Either"A-B=0orsin^(2)A+sin^(2)B-sin^(2)C=0`
`rArr"Either"A=Bor(a^(2))/(k^(2))+(b^(2))/(k^(2))-(c^(2))/(k^(2))=0` [from Eq . (i)]
`therefore` Either A=B or `a^(2)+b^(2)=c^(2)`
Hence , either the triangle is isosceles or it is right angled triangle.
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