Article SIN_COS5, MML version 4.99.1005

:: SIN_COS5:th 1
theorem
for b1 being real set
      st cos b1 <> 0
   holds cosec b1 = (sec b1) / tan b1;

:: SIN_COS5:th 2
theorem
for b1 being real set
      st sin b1 <> 0
   holds cos b1 = (sin b1) * cot b1;

:: SIN_COS5:th 3
theorem
for b1, b2, b3 being real set
      st sin b1 <> 0 & sin b2 <> 0 & sin b3 <> 0
   holds sin ((b1 + b2) + b3) = (((sin b1) * sin b2) * sin b3) * (((((cot b2) * cot b3) + ((cot b1) * cot b3)) + ((cot b1) * cot b2)) - 1);

:: SIN_COS5:th 4
theorem
for b1, b2, b3 being real set
      st sin b1 <> 0 & sin b2 <> 0 & sin b3 <> 0
   holds cos ((b1 + b2) + b3) = - ((((sin b1) * sin b2) * sin b3) * ((((cot b1) + cot b2) + cot b3) - (((cot b1) * cot b2) * cot b3)));

:: SIN_COS5:th 5
theorem
for b1 being real set holds
   sin (2 * b1) = (2 * sin b1) * cos b1;

:: SIN_COS5:th 6
theorem
for b1 being real set
      st cos b1 <> 0
   holds sin (2 * b1) = (2 * tan b1) / (1 + ((tan b1) ^2));

:: SIN_COS5:th 7
theorem
for b1 being real set holds
   cos (2 * b1) = (cos b1) ^2 - ((sin b1) ^2) &
    cos (2 * b1) = (2 * ((cos b1) ^2)) - 1 &
    cos (2 * b1) = 1 - (2 * ((sin b1) ^2));

:: SIN_COS5:th 8
theorem
for b1 being real set
      st cos b1 <> 0
   holds cos (2 * b1) = (1 - ((tan b1) ^2)) / (1 + ((tan b1) ^2));

:: SIN_COS5:th 9
theorem
for b1 being real set
      st cos b1 <> 0
   holds tan (2 * b1) = (2 * tan b1) / (1 - ((tan b1) ^2));

:: SIN_COS5:th 10
theorem
for b1 being real set
      st sin b1 <> 0
   holds cot (2 * b1) = ((cot b1) ^2 - 1) / (2 * cot b1);

:: SIN_COS5:th 11
theorem
for b1 being real set
      st cos b1 <> 0
   holds (sec b1) ^2 = 1 + ((tan b1) ^2);

:: SIN_COS5:th 12
theorem
for b1 being real set holds
   cot b1 = 1 / tan b1;

:: SIN_COS5:th 13
theorem
for b1 being real set
      st cos b1 <> 0 & sin b1 <> 0
   holds sec (2 * b1) = (sec b1) ^2 / (1 - ((tan b1) ^2)) &
    sec (2 * b1) = ((cot b1) + tan b1) / ((cot b1) - tan b1);

:: SIN_COS5:th 14
theorem
for b1 being real set
      st sin b1 <> 0
   holds (cosec b1) ^2 = 1 + ((cot b1) ^2);

:: SIN_COS5:th 15
theorem
for b1 being real set
      st cos b1 <> 0 & sin b1 <> 0
   holds cosec (2 * b1) = ((sec b1) * cosec b1) / 2 &
    cosec (2 * b1) = ((tan b1) + cot b1) / 2;

:: SIN_COS5:th 16
theorem
for b1 being real set holds
   sin (3 * b1) = (- (4 * ((sin b1) |^ 3))) + (3 * sin b1);

:: SIN_COS5:th 17
theorem
for b1 being real set holds
   cos (3 * b1) = (4 * ((cos b1) |^ 3)) - (3 * cos b1);

:: SIN_COS5:th 18
theorem
for b1 being real set
      st cos b1 <> 0
   holds tan (3 * b1) = ((3 * tan b1) - ((tan b1) |^ 3)) / (1 - (3 * ((tan b1) ^2)));

:: SIN_COS5:th 19
theorem
for b1 being real set
      st sin b1 <> 0
   holds cot (3 * b1) = (((cot b1) |^ 3) - (3 * cot b1)) / ((3 * ((cot b1) ^2)) - 1);

:: SIN_COS5:th 20
theorem
for b1 being real set holds
   (sin b1) ^2 = (1 - cos (2 * b1)) / 2;

:: SIN_COS5:th 21
theorem
for b1 being real set holds
   (cos b1) ^2 = (1 + cos (2 * b1)) / 2;

:: SIN_COS5:th 22
theorem
for b1 being real set holds
   (sin b1) |^ 3 = ((3 * sin b1) - sin (3 * b1)) / 4;

:: SIN_COS5:th 23
theorem
for b1 being real set holds
   (cos b1) |^ 3 = ((3 * cos b1) + cos (3 * b1)) / 4;

:: SIN_COS5:th 24
theorem
for b1 being real set holds
   (sin b1) |^ 4 = ((3 - (4 * cos (2 * b1))) + cos (4 * b1)) / 8;

:: SIN_COS5:th 25
theorem
for b1 being real set holds
   (cos b1) |^ 4 = ((3 + (4 * cos (2 * b1))) + cos (4 * b1)) / 8;

:: SIN_COS5:th 26
theorem
for b1 being real set
      st sin (b1 / 2) <> sqrt ((1 - cos b1) / 2)
   holds sin (b1 / 2) = - sqrt ((1 - cos b1) / 2);

:: SIN_COS5:th 27
theorem
for b1 being real set
      st cos (b1 / 2) <> sqrt ((1 + cos b1) / 2)
   holds cos (b1 / 2) = - sqrt ((1 + cos b1) / 2);

:: SIN_COS5:th 28
theorem
for b1 being real set
      st sin (b1 / 2) <> 0
   holds tan (b1 / 2) = (1 - cos b1) / sin b1;

:: SIN_COS5:th 29
theorem
for b1 being real set
      st cos (b1 / 2) <> 0
   holds tan (b1 / 2) = (sin b1) / (1 + cos b1);

:: SIN_COS5:th 30
theorem
for b1 being real set
      st tan (b1 / 2) <> sqrt ((1 - cos b1) / (1 + cos b1))
   holds tan (b1 / 2) = - sqrt ((1 - cos b1) / (1 + cos b1));

:: SIN_COS5:th 31
theorem
for b1 being real set
      st cos (b1 / 2) <> 0
   holds cot (b1 / 2) = (1 + cos b1) / sin b1;

:: SIN_COS5:th 32
theorem
for b1 being real set
      st sin (b1 / 2) <> 0
   holds cot (b1 / 2) = (sin b1) / (1 - cos b1);

:: SIN_COS5:th 33
theorem
for b1 being real set
      st cot (b1 / 2) <> sqrt ((1 + cos b1) / (1 - cos b1))
   holds cot (b1 / 2) = - sqrt ((1 + cos b1) / (1 - cos b1));

:: SIN_COS5:th 34
theorem
for b1 being real set
      st sin (b1 / 2) <> 0 &
         cos (b1 / 2) <> 0 &
         1 - ((tan (b1 / 2)) ^2) <> 0 &
         sec (b1 / 2) <> sqrt ((2 * sec b1) / ((sec b1) + 1))
   holds sec (b1 / 2) = - sqrt ((2 * sec b1) / ((sec b1) + 1));

:: SIN_COS5:th 35
theorem
for b1 being real set
      st sin (b1 / 2) <> 0 &
         cos (b1 / 2) <> 0 &
         1 - ((tan (b1 / 2)) ^2) <> 0 &
         cosec (b1 / 2) <> sqrt ((2 * sec b1) / ((sec b1) - 1))
   holds cosec (b1 / 2) = - sqrt ((2 * sec b1) / ((sec b1) - 1));

:: SIN_COS5:funcnot 1 => SIN_COS5:func 1
definition
  let a1 be real set;
  func coth A1 -> Element of REAL equals
    (cosh a1) / sinh a1;
end;

:: SIN_COS5:def 1
theorem
for b1 being real set holds
   coth b1 = (cosh b1) / sinh b1;

:: SIN_COS5:funcnot 2 => SIN_COS5:func 2
definition
  let a1 be real set;
  func sech A1 -> Element of REAL equals
    1 / cosh a1;
end;

:: SIN_COS5:def 2
theorem
for b1 being real set holds
   sech b1 = 1 / cosh b1;

:: SIN_COS5:funcnot 3 => SIN_COS5:func 3
definition
  let a1 be real set;
  func cosech A1 -> Element of REAL equals
    1 / sinh a1;
end;

:: SIN_COS5:def 3
theorem
for b1 being real set holds
   cosech b1 = 1 / sinh b1;

:: SIN_COS5:th 36
theorem
for b1 being real set holds
   coth b1 = ((exp_R b1) + exp_R - b1) / ((exp_R b1) - exp_R - b1) &
    sech b1 = 2 / ((exp_R b1) + exp_R - b1) &
    cosech b1 = 2 / ((exp_R b1) - exp_R - b1);

:: SIN_COS5:th 37
theorem
for b1 being real set
      st (exp_R b1) - exp_R - b1 <> 0
   holds (tanh b1) * coth b1 = 1;

:: SIN_COS5:th 38
theorem
for b1 being real set holds
   (sech b1) ^2 + ((tanh b1) ^2) = 1;

:: SIN_COS5:th 39
theorem
for b1 being real set
      st sinh b1 <> 0
   holds (coth b1) ^2 - ((cosech b1) ^2) = 1;

:: SIN_COS5:th 40
theorem
for b1, b2 being real set
      st sinh b1 <> 0 & sinh b2 <> 0
   holds coth (b1 + b2) = (1 + ((coth b1) * coth b2)) / ((coth b1) + coth b2);

:: SIN_COS5:th 41
theorem
for b1, b2 being real set
      st sinh b1 <> 0 & sinh b2 <> 0
   holds coth (b1 - b2) = (1 - ((coth b1) * coth b2)) / ((coth b1) - coth b2);

:: SIN_COS5:th 42
theorem
for b1, b2 being real set
      st sinh b1 <> 0 & sinh b2 <> 0
   holds (coth b1) + coth b2 = (sinh (b1 + b2)) / ((sinh b1) * sinh b2) &
    (coth b1) - coth b2 = - ((sinh (b1 - b2)) / ((sinh b1) * sinh b2));

:: SIN_COS5:th 43
theorem
for b1 being real set holds
   sinh (3 * b1) = (3 * sinh b1) + (4 * ((sinh b1) |^ 3));

:: SIN_COS5:th 44
theorem
for b1 being real set holds
   cosh (3 * b1) = (4 * ((cosh b1) |^ 3)) - (3 * cosh b1);

:: SIN_COS5:th 45
theorem
for b1 being real set
      st sinh b1 <> 0
   holds coth (2 * b1) = (1 + ((coth b1) ^2)) / (2 * coth b1);

:: SIN_COS5:th 46
theorem
for b1 being real set
      st 0 <= b1
   holds 0 <= sinh b1;

:: SIN_COS5:th 47
theorem
for b1 being real set
      st b1 <= 0
   holds sinh b1 <= 0;

:: SIN_COS5:th 48
theorem
for b1 being real set holds
   cosh (b1 / 2) = sqrt (((cosh b1) + 1) / 2);

:: SIN_COS5:th 49
theorem
for b1 being real set
      st sinh (b1 / 2) <> 0
   holds tanh (b1 / 2) = ((cosh b1) - 1) / sinh b1;

:: SIN_COS5:th 50
theorem
for b1 being real set
      st cosh (b1 / 2) <> 0
   holds tanh (b1 / 2) = (sinh b1) / ((cosh b1) + 1);

:: SIN_COS5:th 51
theorem
for b1 being real set
      st sinh (b1 / 2) <> 0
   holds coth (b1 / 2) = (sinh b1) / ((cosh b1) - 1);

:: SIN_COS5:th 52
theorem
for b1 being real set
      st cosh (b1 / 2) <> 0
   holds coth (b1 / 2) = ((cosh b1) + 1) / sinh b1;