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;