Article SIN_COS4, MML version 4.99.1005

:: SIN_COS4:funcnot 1 => SIN_COS4:func 1
definition
  let a1 be real set;
  func tan A1 -> Element of REAL equals
    (sin a1) / cos a1;
end;

:: SIN_COS4:def 1
theorem
for b1 being real set holds
   tan b1 = (sin b1) / cos b1;

:: SIN_COS4:funcnot 2 => SIN_COS4:func 2
definition
  let a1 be real set;
  func cot A1 -> Element of REAL equals
    (cos a1) / sin a1;
end;

:: SIN_COS4:def 2
theorem
for b1 being real set holds
   cot b1 = (cos b1) / sin b1;

:: SIN_COS4:funcnot 3 => SIN_COS4:func 3
definition
  let a1 be real set;
  func cosec A1 -> Element of REAL equals
    1 / sin a1;
end;

:: SIN_COS4:def 3
theorem
for b1 being real set holds
   cosec b1 = 1 / sin b1;

:: SIN_COS4:funcnot 4 => SIN_COS4:func 4
definition
  let a1 be real set;
  func sec A1 -> Element of REAL equals
    1 / cos a1;
end;

:: SIN_COS4:def 4
theorem
for b1 being real set holds
   sec b1 = 1 / cos b1;

:: SIN_COS4:th 2
theorem
for b1 being real set holds
   tan - b1 = - tan b1;

:: SIN_COS4:th 3
theorem
for b1 being real set holds
   cosec - b1 = - (1 / sin b1);

:: SIN_COS4:th 4
theorem
for b1 being real set holds
   cot - b1 = - cot b1;

:: SIN_COS4:th 6
theorem
for b1 being real set holds
   (sin b1) * sin b1 = 1 - ((cos b1) * cos b1);

:: SIN_COS4:th 7
theorem
for b1 being real set holds
   (cos b1) * cos b1 = 1 - ((sin b1) * sin b1);

:: SIN_COS4:th 8
theorem
for b1 being real set
      st cos b1 <> 0
   holds sin b1 = (cos b1) * tan b1;

:: SIN_COS4:th 11
theorem
for b1, b2 being real set
      st cos b1 <> 0 & cos b2 <> 0
   holds tan (b1 + b2) = ((tan b1) + tan b2) / (1 - ((tan b1) * tan b2));

:: SIN_COS4:th 12
theorem
for b1, b2 being real set
      st cos b1 <> 0 & cos b2 <> 0
   holds tan (b1 - b2) = ((tan b1) - tan b2) / (1 + ((tan b1) * tan b2));

:: SIN_COS4:th 13
theorem
for b1, b2 being real set
      st sin b1 <> 0 & sin b2 <> 0
   holds cot (b1 + b2) = (((cot b1) * cot b2) - 1) / ((cot b2) + cot b1);

:: SIN_COS4:th 14
theorem
for b1, b2 being real set
      st sin b1 <> 0 & sin b2 <> 0
   holds cot (b1 - b2) = (((cot b1) * cot b2) + 1) / ((cot b2) - cot b1);

:: SIN_COS4:th 15
theorem
for b1, b2, b3 being real set
      st cos b1 <> 0 & cos b2 <> 0 & cos b3 <> 0
   holds sin ((b1 + b2) + b3) = (((cos b1) * cos b2) * cos b3) * ((((tan b1) + tan b2) + tan b3) - (((tan b1) * tan b2) * tan b3));

:: SIN_COS4:th 16
theorem
for b1, b2, b3 being real set
      st cos b1 <> 0 & cos b2 <> 0 & cos b3 <> 0
   holds cos ((b1 + b2) + b3) = (((cos b1) * cos b2) * cos b3) * (((1 - ((tan b2) * tan b3)) - ((tan b3) * tan b1)) - ((tan b1) * tan b2));

:: SIN_COS4:th 17
theorem
for b1, b2, b3 being real set
      st cos b1 <> 0 & cos b2 <> 0 & cos b3 <> 0
   holds tan ((b1 + b2) + b3) = ((((tan b1) + tan b2) + tan b3) - (((tan b1) * tan b2) * tan b3)) / (((1 - ((tan b2) * tan b3)) - ((tan b3) * tan b1)) - ((tan b1) * tan b2));

:: SIN_COS4:th 18
theorem
for b1, b2, b3 being real set
      st sin b1 <> 0 & sin b2 <> 0 & sin b3 <> 0
   holds cot ((b1 + b2) + b3) = ((((((cot b1) * cot b2) * cot b3) - cot b1) - cot b2) - cot b3) / (((((cot b2) * cot b3) + ((cot b3) * cot b1)) + ((cot b1) * cot b2)) - 1);

:: SIN_COS4:th 19
theorem
for b1, b2 being real set holds
(sin b1) + sin b2 = 2 * ((cos ((b1 - b2) / 2)) * sin ((b1 + b2) / 2));

:: SIN_COS4:th 20
theorem
for b1, b2 being real set holds
(sin b1) - sin b2 = 2 * ((cos ((b1 + b2) / 2)) * sin ((b1 - b2) / 2));

:: SIN_COS4:th 21
theorem
for b1, b2 being real set holds
(cos b1) + cos b2 = 2 * ((cos ((b1 + b2) / 2)) * cos ((b1 - b2) / 2));

:: SIN_COS4:th 22
theorem
for b1, b2 being real set holds
(cos b1) - cos b2 = - (2 * ((sin ((b1 + b2) / 2)) * sin ((b1 - b2) / 2)));

:: SIN_COS4:th 23
theorem
for b1, b2 being real set
      st cos b1 <> 0 & cos b2 <> 0
   holds (tan b1) + tan b2 = (sin (b1 + b2)) / ((cos b1) * cos b2);

:: SIN_COS4:th 24
theorem
for b1, b2 being real set
      st cos b1 <> 0 & cos b2 <> 0
   holds (tan b1) - tan b2 = (sin (b1 - b2)) / ((cos b1) * cos b2);

:: SIN_COS4:th 25
theorem
for b1, b2 being real set
      st cos b1 <> 0 & sin b2 <> 0
   holds (tan b1) + cot b2 = (cos (b1 - b2)) / ((cos b1) * sin b2);

:: SIN_COS4:th 26
theorem
for b1, b2 being real set
      st cos b1 <> 0 & sin b2 <> 0
   holds (tan b1) - cot b2 = - ((cos (b1 + b2)) / ((cos b1) * sin b2));

:: SIN_COS4:th 27
theorem
for b1, b2 being real set
      st sin b1 <> 0 & sin b2 <> 0
   holds (cot b1) + cot b2 = (sin (b1 + b2)) / ((sin b1) * sin b2);

:: SIN_COS4:th 28
theorem
for b1, b2 being real set
      st sin b1 <> 0 & sin b2 <> 0
   holds (cot b1) - cot b2 = - ((sin (b1 - b2)) / ((sin b1) * sin b2));

:: SIN_COS4:th 29
theorem
for b1, b2 being real set holds
(sin (b1 + b2)) + sin (b1 - b2) = 2 * ((sin b1) * cos b2);

:: SIN_COS4:th 30
theorem
for b1, b2 being real set holds
(sin (b1 + b2)) - sin (b1 - b2) = 2 * ((cos b1) * sin b2);

:: SIN_COS4:th 31
theorem
for b1, b2 being real set holds
(cos (b1 + b2)) + cos (b1 - b2) = 2 * ((cos b1) * cos b2);

:: SIN_COS4:th 32
theorem
for b1, b2 being real set holds
(cos (b1 + b2)) - cos (b1 - b2) = - (2 * ((sin b1) * sin b2));

:: SIN_COS4:th 33
theorem
for b1, b2 being real set holds
(sin b1) * sin b2 = - ((1 / 2) * ((cos (b1 + b2)) - cos (b1 - b2)));

:: SIN_COS4:th 34
theorem
for b1, b2 being real set holds
(sin b1) * cos b2 = (1 / 2) * ((sin (b1 + b2)) + sin (b1 - b2));

:: SIN_COS4:th 35
theorem
for b1, b2 being real set holds
(cos b1) * sin b2 = (1 / 2) * ((sin (b1 + b2)) - sin (b1 - b2));

:: SIN_COS4:th 36
theorem
for b1, b2 being real set holds
(cos b1) * cos b2 = (1 / 2) * ((cos (b1 + b2)) + cos (b1 - b2));

:: SIN_COS4:th 37
theorem
for b1, b2, b3 being real set holds
((sin b1) * sin b2) * sin b3 = (1 / 4) * ((((sin ((b1 + b2) - b3)) + sin ((b2 + b3) - b1)) + sin ((b3 + b1) - b2)) - sin ((b1 + b2) + b3));

:: SIN_COS4:th 38
theorem
for b1, b2, b3 being real set holds
((sin b1) * sin b2) * cos b3 = (1 / 4) * ((((- cos ((b1 + b2) - b3)) + cos ((b2 + b3) - b1)) + cos ((b3 + b1) - b2)) - cos ((b1 + b2) + b3));

:: SIN_COS4:th 39
theorem
for b1, b2, b3 being real set holds
((sin b1) * cos b2) * cos b3 = (1 / 4) * ((((sin ((b1 + b2) - b3)) - sin ((b2 + b3) - b1)) + sin ((b3 + b1) - b2)) + sin ((b1 + b2) + b3));

:: SIN_COS4:th 40
theorem
for b1, b2, b3 being real set holds
((cos b1) * cos b2) * cos b3 = (1 / 4) * ((((cos ((b1 + b2) - b3)) + cos ((b2 + b3) - b1)) + cos ((b3 + b1) - b2)) + cos ((b1 + b2) + b3));

:: SIN_COS4:th 41
theorem
for b1, b2 being real set holds
(sin (b1 + b2)) * sin (b1 - b2) = ((sin b1) * sin b1) - ((sin b2) * sin b2);

:: SIN_COS4:th 42
theorem
for b1, b2 being real set holds
(sin (b1 + b2)) * sin (b1 - b2) = ((cos b2) * cos b2) - ((cos b1) * cos b1);

:: SIN_COS4:th 43
theorem
for b1, b2 being real set holds
(sin (b1 + b2)) * cos (b1 - b2) = ((sin b1) * cos b1) + ((sin b2) * cos b2);

:: SIN_COS4:th 44
theorem
for b1, b2 being real set holds
(cos (b1 + b2)) * sin (b1 - b2) = ((sin b1) * cos b1) - ((sin b2) * cos b2);

:: SIN_COS4:th 45
theorem
for b1, b2 being real set holds
(cos (b1 + b2)) * cos (b1 - b2) = ((cos b1) * cos b1) - ((sin b2) * sin b2);

:: SIN_COS4:th 46
theorem
for b1, b2 being real set holds
(cos (b1 + b2)) * cos (b1 - b2) = ((cos b2) * cos b2) - ((sin b1) * sin b1);

:: SIN_COS4:th 47
theorem
for b1, b2 being real set
      st cos b1 <> 0 & cos b2 <> 0
   holds (sin (b1 + b2)) / sin (b1 - b2) = ((tan b1) + tan b2) / ((tan b1) - tan b2);

:: SIN_COS4:th 48
theorem
for b1, b2 being real set
      st cos b1 <> 0 & cos b2 <> 0
   holds (cos (b1 + b2)) / cos (b1 - b2) = (1 - ((tan b1) * tan b2)) / (1 + ((tan b1) * tan b2));

:: SIN_COS4:th 49
theorem
for b1, b2 being real set holds
((sin b1) + sin b2) / ((sin b1) - sin b2) = (tan ((b1 + b2) / 2)) * cot ((b1 - b2) / 2);

:: SIN_COS4:th 50
theorem
for b1, b2 being real set
      st cos ((b1 - b2) / 2) <> 0
   holds ((sin b1) + sin b2) / ((cos b1) + cos b2) = tan ((b1 + b2) / 2);

:: SIN_COS4:th 51
theorem
for b1, b2 being real set
      st cos ((b1 + b2) / 2) <> 0
   holds ((sin b1) - sin b2) / ((cos b1) + cos b2) = tan ((b1 - b2) / 2);

:: SIN_COS4:th 52
theorem
for b1, b2 being real set
      st sin ((b1 + b2) / 2) <> 0
   holds ((sin b1) + sin b2) / ((cos b2) - cos b1) = cot ((b1 - b2) / 2);

:: SIN_COS4:th 53
theorem
for b1, b2 being real set
      st sin ((b1 - b2) / 2) <> 0
   holds ((sin b1) - sin b2) / ((cos b2) - cos b1) = cot ((b1 + b2) / 2);

:: SIN_COS4:th 54
theorem
for b1, b2 being real set holds
((cos b1) + cos b2) / ((cos b1) - cos b2) = (cot ((b1 + b2) / 2)) * cot ((b2 - b1) / 2);