Expression 40: cosine left parenthesis, "a" plus 45 , right parenthesis equals sine left parenthesis, 45 minus "a" , right parenthesis equals StartFraction, StartRoot, 2 , EndRoot Over 2 , EndFraction left parenthesis, cos "a" minus sin "a" , right parenthesis equals StartFraction, StartRoot, 2 , EndRoot Over 2 , EndFraction left parenthesis, StartRoot, 1 minus sin squared "a" , EndRoot minus sin "a" , right parenthesiscosa+45=sin45−a=22cosa−sina=221−sin2a−sina
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Expression 41: StartFraction, 4 plus 2sin "a" Over 4 plus 2 StartRoot, 1 minus sin squared "a" , EndRoot , EndFraction sine "a" plus StartRoot, 1 minus sin squared "a" , EndRoot equals StartRoot, 2 , EndRoot left parenthesis, StartFraction, 4 plus 2 StartRoot, 2 , EndRoot StartNestedFraction, StartRoot, 2 , EndRoot NestedOver 2 , EndNestedFraction left parenthesis, sin "a" plus StartRoot, 1 minus sin squared "a" , EndRoot , right parenthesis Over 4 plus 2 StartRoot, 2 , EndRoot StartNestedFraction, StartRoot, 2 , EndRoot NestedOver 2 , EndNestedFraction left parenthesis, StartRoot, 1 minus sin squared "a" , EndRoot minus sin "a" , right parenthesis , EndFraction StartFraction, StartRoot, 2 , EndRoot Over 2 , EndFraction left parenthesis, StartRoot, 1 minus sin squared "a" , EndRoot minus sin "a" , right parenthesis plus StartFraction, StartRoot, 2 , EndRoot Over 2 , EndFraction left parenthesis, sin "a" plus StartRoot, 1 minus sin squared "a" , EndRoot , right parenthesis , right parenthesis4+2sina4+21−sin2asina+1−sin2a=24+2222sina+1−sin2a4+22221−sin2a−sina221−sin2a−sina+22sina+1−sin2a
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Rewrite with b = sin a
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Expression 43: StartFraction, 4 plus 2 "b" Over 4 plus 2 StartRoot, 1 minus "b" squared , EndRoot , EndFraction "b" plus StartRoot, 1 minus "b" squared , EndRoot equals StartRoot, 2 , EndRoot left parenthesis, StartFraction, 4 plus 2 StartRoot, 2 , EndRoot StartNestedFraction, StartRoot, 2 , EndRoot NestedOver 2 , EndNestedFraction left parenthesis, "b" plus StartRoot, 1 minus "b" squared , EndRoot , right parenthesis Over 4 plus 2 StartRoot, 2 , EndRoot StartNestedFraction, StartRoot, 2 , EndRoot NestedOver 2 , EndNestedFraction left parenthesis, StartRoot, 1 minus "b" squared , EndRoot minus "b" , right parenthesis , EndFraction StartFraction, StartRoot, 2 , EndRoot Over 2 , EndFraction left parenthesis, StartRoot, 1 minus "b" squared , EndRoot minus "b" , right parenthesis plus StartFraction, StartRoot, 2 , EndRoot Over 2 , EndFraction left parenthesis, "b" plus StartRoot, 1 minus "b" squared , EndRoot , right parenthesis , right parenthesis4+2b4+21−b2b+1−b2=24+2222b+1−b24+22221−b2−b221−b2−b+22b+1−b2
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Expression 44: StartFraction, 4 plus 2 "b" Over 4 plus 2 StartRoot, 1 minus "b" squared , EndRoot , EndFraction "b" plus StartRoot, 1 minus "b" squared , EndRoot equals left parenthesis, StartFraction, 4 plus 2 StartRoot, 2 , EndRoot StartNestedFraction, StartRoot, 2 , EndRoot NestedOver 2 , EndNestedFraction left parenthesis, "b" plus StartRoot, 1 minus "b" squared , EndRoot , right parenthesis Over 4 plus 2 StartRoot, 2 , EndRoot StartNestedFraction, StartRoot, 2 , EndRoot NestedOver 2 , EndNestedFraction left parenthesis, StartRoot, 1 minus "b" squared , EndRoot minus "b" , right parenthesis , EndFraction left parenthesis, StartRoot, 1 minus "b" squared , EndRoot minus "b" , right parenthesis plus left parenthesis, "b" plus StartRoot, 1 minus "b" squared , EndRoot , right parenthesis , right parenthesis4+2b4+21−b2b+1−b2=4+2222b+1−b24+22221−b2−b1−b2−b+b+1−b2
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Expression 45: StartFraction, 4 plus 2 "b" Over 4 plus 2 StartRoot, 1 minus "b" squared , EndRoot , EndFraction "b" plus StartRoot, 1 minus "b" squared , EndRoot equals left parenthesis, StartFraction, 4 plus 2 left parenthesis, "b" plus StartRoot, 1 minus "b" squared , EndRoot , right parenthesis Over 4 plus 2 left parenthesis, StartRoot, 1 minus "b" squared , EndRoot minus "b" , right parenthesis , EndFraction left parenthesis, StartRoot, 1 minus "b" squared , EndRoot minus "b" , right parenthesis plus left parenthesis, "b" plus StartRoot, 1 minus "b" squared , EndRoot , right parenthesis , right parenthesis4+2b4+21−b2b+1−b2=4+2b+1−b24+21−b2−b1−b2−b+b+1−b2
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Expression 46: StartFraction, 2 plus "b" Over 2 plus StartRoot, 1 minus "b" squared , EndRoot , EndFraction "b" plus StartRoot, 1 minus "b" squared , EndRoot equals left parenthesis, StartFraction, 2 plus "b" plus StartRoot, 1 minus "b" squared , EndRoot Over 2 minus "b" plus StartRoot, 1 minus "b" squared , EndRoot , EndFraction left parenthesis, StartRoot, 1 minus "b" squared , EndRoot minus "b" , right parenthesis plus left parenthesis, "b" plus StartRoot, 1 minus "b" squared , EndRoot , right parenthesis , right parenthesis2+b2+1−b2b+1−b2=2+b+1−b22−b+1−b21−b2−b+b+1−b2
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Expression 47: StartFraction, 2 plus "b" Over 2 plus StartRoot, 1 minus "b" squared , EndRoot , EndFraction "b" plus StartRoot, 1 minus "b" squared , EndRoot equals left parenthesis, left parenthesis, 1 minus StartFraction, 2 plus "b" plus StartRoot, 1 minus "b" squared , EndRoot Over 2 minus "b" plus StartRoot, 1 minus "b" squared , EndRoot , EndFraction , right parenthesis "b" plus left parenthesis, 1 plus StartFraction, 2 plus "b" plus StartRoot, 1 minus "b" squared , EndRoot Over 2 minus "b" plus StartRoot, 1 minus "b" squared , EndRoot , EndFraction , right parenthesis StartRoot, 1 minus "b" squared , EndRoot , right parenthesis2+b2+1−b2b+1−b2=1−2+b+1−b22−b+1−b2b+1+2+b+1−b22−b+1−b21−b2
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Expression 48: StartFraction, 2 plus "b" Over 2 plus StartRoot, 1 minus "b" squared , EndRoot , EndFraction "b" plus StartRoot, 1 minus "b" squared , EndRoot equals left parenthesis, 1 minus StartFraction, 2 plus "b" plus StartRoot, 1 minus "b" squared , EndRoot Over 2 minus "b" plus StartRoot, 1 minus "b" squared , EndRoot , EndFraction , right parenthesis left parenthesis, "b" minus StartRoot, 1 minus "b" squared , EndRoot , right parenthesis plus 2 StartRoot, 1 minus "b" squared , EndRoot2+b2+1−b2b+1−b2=1−2+b+1−b22−b+1−b2b−1−b2+21−b2
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Expression 49: StartFraction, 2 plus "b" Over 2 plus StartRoot, 1 minus "b" squared , EndRoot , EndFraction "b" minus StartRoot, 1 minus "b" squared , EndRoot equals left parenthesis, 1 minus StartFraction, 2 plus "b" plus StartRoot, 1 minus "b" squared , EndRoot Over 2 minus "b" plus StartRoot, 1 minus "b" squared , EndRoot , EndFraction , right parenthesis left parenthesis, "b" minus StartRoot, 1 minus "b" squared , EndRoot , right parenthesis2+b2+1−b2b−1−b2=1−2+b+1−b22−b+1−b2b−1−b2
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Expression 50: left parenthesis, 2 plus "b" , right parenthesis "b" minus left parenthesis, 2 plus StartRoot, 1 minus "b" squared , EndRoot , right parenthesis StartRoot, 1 minus "b" squared , EndRoot equals left parenthesis, 1 minus StartFraction, 2 plus "b" plus StartRoot, 1 minus "b" squared , EndRoot Over 2 minus "b" plus StartRoot, 1 minus "b" squared , EndRoot , EndFraction , right parenthesis left parenthesis, "b" minus StartRoot, 1 minus "b" squared , EndRoot , right parenthesis left parenthesis, 2 plus StartRoot, 1 minus "b" squared , EndRoot , right parenthesis2+bb−2+1−b21−b2=1−2+b+1−b22−b+1−b2b−1−b22+1−b2
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Expression 51: left parenthesis, 2 plus "b" , right parenthesis "b" minus 2 StartRoot, 1 minus "b" squared , EndRoot minus 1 plus "b" squared equals left parenthesis, 1 minus StartFraction, 2 plus "b" plus StartRoot, 1 minus "b" squared , EndRoot Over 2 minus "b" plus StartRoot, 1 minus "b" squared , EndRoot , EndFraction , right parenthesis left parenthesis, "b" minus StartRoot, 1 minus "b" squared , EndRoot , right parenthesis left parenthesis, 2 plus StartRoot, 1 minus "b" squared , EndRoot , right parenthesis2+bb−21−b2−1+b2=1−2+b+1−b22−b+1−b2b−1−b22+1−b2
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Expression 52: left parenthesis, left parenthesis, 2 plus "b" , right parenthesis "b" minus 2 StartRoot, 1 minus "b" squared , EndRoot minus 1 plus "b" squared , right parenthesis left parenthesis, 2 minus "b" plus StartRoot, 1 minus "b" squared , EndRoot , right parenthesis equals left parenthesis, 2 minus "b" plus StartRoot, 1 minus "b" squared , EndRoot minus left parenthesis, 2 plus "b" plus StartRoot, 1 minus "b" squared , EndRoot , right parenthesis , right parenthesis left parenthesis, "b" minus StartRoot, 1 minus "b" squared , EndRoot , right parenthesis left parenthesis, 2 plus StartRoot, 1 minus "b" squared , EndRoot , right parenthesis2+bb−21−b2−1+b22−b+1−b2=2−b+1−b2−2+b+1−b2b−1−b22+1−b2
Expression 56: "b" squared plus "c" squared equals 1b2+c2=1
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Expression 57: left parenthesis, left parenthesis, 2 plus "b" , right parenthesis "b" minus 2 "c" minus 1 plus "b" squared , right parenthesis left parenthesis, 2 minus "b" plus "c" , right parenthesis equals left parenthesis, 2 minus "b" plus "c" minus left parenthesis, 2 plus "b" plus "c" , right parenthesis , right parenthesis left parenthesis, "b" minus "c" , right parenthesis left parenthesis, 2 plus "c" , right parenthesis2+bb−2c−1+b22−b+c=2−b+c−2+b+cb−c2+c
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Expression 58: left parenthesis, 2 "b" plus "b" squared minus 2 "c" minus 1 plus "b" squared , right parenthesis left parenthesis, 2 minus "b" plus "c" , right parenthesis equals left parenthesis, 2 minus "b" plus "c" minus left parenthesis, 2 plus "b" plus "c" , right parenthesis , right parenthesis left parenthesis, "b" minus "c" , right parenthesis left parenthesis, 2 plus "c" , right parenthesis2b+b2−2c−1+b22−b+c=2−b+c−2+b+cb−c2+c
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Expression 59: 4 "b" plus 4 "b" squared minus 4 "c" minus 2 minus 2 "b" squared minus 2 "b" cubed plus 2 "b" "c" plus "b" plus 2 "b" "c" plus 2 "b" squared "c" minus 2 "c" squared minus "c" equals left parenthesis, 2 minus "b" plus "c" minus 2 plus "b" plus "c" , right parenthesis left parenthesis, "b" minus "c" , right parenthesis left parenthesis, 2 plus "c" , right parenthesis4b+4b2−4c−2−2b2−2b3+2bc+b+2bc+2b2c−2c2−c=2−b+c−2+b+cb−c2+c
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Expression 60: 4 "b" plus 4 "b" squared minus 4 "c" minus 2 minus 2 "b" squared minus 2 "b" cubed plus 2 "b" "c" plus "b" plus 2 "b" "c" plus 2 "b" squared "c" minus 2 "c" squared minus "c" equals 2 "c" left parenthesis, "b" minus "c" , right parenthesis left parenthesis, 2 plus "c" , right parenthesis4b+4b2−4c−2−2b2−2b3+2bc+b+2bc+2b2c−2c2−c=2cb−c2+c
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Expression 61: 4 "b" plus 4 "b" squared minus 4 "c" minus 2 minus 2 "b" squared minus 2 "b" cubed plus 2 "b" "c" plus "b" plus 2 "b" "c" plus 2 "b" squared "c" minus 2 "c" squared minus "c" equals 2 "c" left parenthesis, 2 "b" plus "b" "c" minus 2 "c" minus "c" squared , right parenthesis4b+4b2−4c−2−2b2−2b3+2bc+b+2bc+2b2c−2c2−c=2c2b+bc−2c−c2
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Expression 62: 4 "b" plus 4 "b" squared minus 4 "c" minus 2 minus 2 "b" squared minus 2 "b" cubed plus 2 "b" "c" plus "b" plus 2 "b" "c" plus 2 "b" squared "c" minus 2 "c" squared minus "c" equals 4 "b" "c" plus 2 "b" "c" squared minus 4 "c" squared minus 2 "c" cubed4b+4b2−4c−2−2b2−2b3+2bc+b+2bc+2b2c−2c2−c=4bc+2bc2−4c2−2c3
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Expression 63: 5 "b" plus 2 "b" squared minus 5 "c" minus 2 minus 2 "b" cubed plus 2 "b" squared "c" plus 2 "c" squared equals 2 "b" "c" squared minus 2 "c" cubed5b+2b2−5c−2−2b3+2b2c+2c2=2bc2−2c3
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Expression 64: 5 "b" minus 5 "c" minus 2 "b" cubed plus 2 "b" squared "c" equals 2 "b" "c" squared minus 2 "c" cubed5b−5c−2b3+2b2c=2bc2−2c3
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Expression 65: 5 "b" minus 5 "c" minus 2 "b" cubed minus 2 "b" "c" squared plus 2 "b" squared "c" plus 2 "c" cubed equals 05b−5c−2b3−2bc2+2b2c+2c3=0
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Expression 66: 5 "b" minus 5 "c" minus 2 "b" left parenthesis, "b" squared plus "c" squared , right parenthesis plus 2 "c" left parenthesis, "b" squared plus "c" squared , right parenthesis equals 05b−5c−2bb2+c2+2cb2+c2=0
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Expression 67: "b" squared plus "c" squared equals 1b2+c2=1
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Expression 68: 5 "b" minus 5 "c" minus 2 "b" plus 2 "c" equals 05b−5c−2b+2c=0
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Expression 69: 3 "b" minus 3 "c" equals 03b−3c=0
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Expression 70: 3 "b" equals 3 "c"3b=3c
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b=c
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Expression 72: "b" squared plus "c" squared equals 1b2+c2=1