SOLUTION: Suppose that an isosceles triangle has sides of length a, a, and b and is such that cos(2A)=sin(B). Show that sin^2(B)=-2sin(A)cos(A)cos(B).

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Question 35998: Suppose that an isosceles triangle has sides of length a, a, and b and is such that cos(2A)=sin(B). Show that sin^2(B)=-2sin(A)cos(A)cos(B).
Answer by venugopalramana(3286) About Me  (Show Source):
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suppose that an isosceles triangle has sides of length a, a, and b and is
> such that cos(2A)=sin(B). Show that sin^2(B)=-2sin(A)cos(A)cos(B).
COS(2A)=SIN(B).............................................................I
IN A TRIANGLE
A+B+C=180BUT SINCE C=A
B+2A=180...................................................................II
2A=180-B
COS(2A)=COS(180-B)=-COS(B)....FROM EQN.I,WE GET
-COS(B)=SIN(B)....OR......COS(B)=-SIN(B)..................................III
WE HAVE T.S.T.
SIN^2B=-2SIN(A)COS(A)COS(B),SUBSTITUTING FROM III IN RHS
TST....SIN^2B=-2SIN(A)COS(A)(-SIN(B))....OR.......
TST....SIN(B)=2SIN(A)COS(A)=SIN(2A)
USING EQN.II..............2A=180-B.....HENCE
SIN(2A)=SIN(180-B)=SIN(B)......THUS WE PROVED THE RESULT.