Article SIN_COS8, MML version 4.99.1005
:: SIN_COS8:th 1
theorem
for b1 being real set holds
tanh b1 = (sinh b1) / cosh b1 & tanh 0 = 0;
:: SIN_COS8:th 2
theorem
for b1 being real set holds
sinh b1 = 1 / cosech b1 & cosh b1 = 1 / sech b1 & tanh b1 = 1 / coth b1;
:: SIN_COS8:th 3
theorem
for b1 being real set holds
sech b1 <= 1 & 0 < sech b1 & sech 0 = 1;
:: SIN_COS8:th 4
theorem
for b1 being real set
st 0 <= b1
holds 0 <= tanh b1;
:: SIN_COS8:th 5
theorem
for b1 being real set holds
cosh b1 = 1 / sqrt (1 - ((tanh b1) ^2)) &
sinh b1 = (tanh b1) / sqrt (1 - ((tanh b1) ^2));
:: SIN_COS8:th 6
theorem
for b1 being real set
for b2 being Element of NAT holds
((cosh b1) + sinh b1) |^ b2 = (cosh (b2 * b1)) + sinh (b2 * b1) &
((cosh b1) - sinh b1) |^ b2 = (cosh (b2 * b1)) - sinh (b2 * b1);
:: SIN_COS8:th 7
theorem
for b1 being real set holds
exp_R b1 = (cosh b1) + sinh b1 &
exp_R - b1 = (cosh b1) - sinh b1 &
exp_R b1 = ((cosh (b1 / 2)) + sinh (b1 / 2)) / ((cosh (b1 / 2)) - sinh (b1 / 2)) &
exp_R - b1 = ((cosh (b1 / 2)) - sinh (b1 / 2)) / ((cosh (b1 / 2)) + sinh (b1 / 2)) &
exp_R b1 = (1 + tanh (b1 / 2)) / (1 - tanh (b1 / 2)) &
exp_R - b1 = (1 - tanh (b1 / 2)) / (1 + tanh (b1 / 2));
:: SIN_COS8:th 8
theorem
for b1 being real set
st b1 <> 0
holds exp_R b1 = ((coth (b1 / 2)) + 1) / ((coth (b1 / 2)) - 1) &
exp_R - b1 = ((coth (b1 / 2)) - 1) / ((coth (b1 / 2)) + 1);
:: SIN_COS8:th 9
theorem
for b1 being real set holds
((cosh b1) + sinh b1) / ((cosh b1) - sinh b1) = (1 + tanh b1) / (1 - tanh b1);
:: SIN_COS8:th 10
theorem
for b1, b2 being real set
st b1 <> 0
holds (coth b1) + tanh b2 = (cosh (b1 + b2)) / ((sinh b1) * cosh b2) &
(coth b1) - tanh b2 = (cosh (b1 - b2)) / ((sinh b1) * cosh b2);
:: SIN_COS8:th 11
theorem
for b1, b2 being real set holds
(sinh b1) * sinh b2 = (1 / 2) * ((cosh (b1 + b2)) - cosh (b1 - b2)) &
(sinh b1) * cosh b2 = (1 / 2) * ((sinh (b1 + b2)) + sinh (b1 - b2)) &
(cosh b1) * sinh b2 = (1 / 2) * ((sinh (b1 + b2)) - sinh (b1 - b2)) &
(cosh b1) * cosh b2 = (1 / 2) * ((cosh (b1 + b2)) + cosh (b1 - b2));
:: SIN_COS8:th 12
theorem
for b1, b2 being real set holds
(sinh b1) ^2 - ((cosh b2) ^2) = ((sinh (b1 + b2)) * sinh (b1 - b2)) - 1;
:: SIN_COS8:th 13
theorem
for b1, b2 being real set holds
((sinh b1) - sinh b2) ^2 - (((cosh b1) - cosh b2) ^2) = 4 * ((sinh ((b1 - b2) / 2)) ^2) &
((cosh b1) + cosh b2) ^2 - (((sinh b1) + sinh b2) ^2) = 4 * ((cosh ((b1 - b2) / 2)) ^2);
:: SIN_COS8:th 14
theorem
for b1, b2 being real set holds
((sinh b1) + sinh b2) / ((sinh b1) - sinh b2) = (tanh ((b1 + b2) / 2)) * coth ((b1 - b2) / 2);
:: SIN_COS8:th 15
theorem
for b1, b2 being real set holds
((cosh b1) + cosh b2) / ((cosh b1) - cosh b2) = (coth ((b1 + b2) / 2)) * coth ((b1 - b2) / 2);
:: SIN_COS8:th 16
theorem
for b1, b2 being real set
st b1 - b2 <> 0
holds ((sinh b1) + sinh b2) / ((cosh b1) + cosh b2) = ((cosh b1) - cosh b2) / ((sinh b1) - sinh b2);
:: SIN_COS8:th 17
theorem
for b1, b2 being real set
st b1 + b2 <> 0
holds ((sinh b1) - sinh b2) / ((cosh b1) + cosh b2) = ((cosh b1) - cosh b2) / ((sinh b1) + sinh b2);
:: SIN_COS8:th 18
theorem
for b1, b2 being real set holds
((sinh b1) + sinh b2) / ((cosh b1) + cosh b2) = tanh ((b1 / 2) + (b2 / 2)) &
((sinh b1) - sinh b2) / ((cosh b1) + cosh b2) = tanh ((b1 / 2) - (b2 / 2));
:: SIN_COS8:th 19
theorem
for b1, b2 being real set holds
((tanh b1) + tanh b2) / ((tanh b1) - tanh b2) = (sinh (b1 + b2)) / sinh (b1 - b2);
:: SIN_COS8:th 20
theorem
for b1, b2 being real set holds
(((sinh (b1 - b2)) + sinh b1) + sinh (b1 + b2)) / (((cosh (b1 - b2)) + cosh b1) + cosh (b1 + b2)) = tanh b1;
:: SIN_COS8:th 21
theorem
for b1, b2, b3 being real set holds
sinh ((b1 + b2) + b3) = ((((((tanh b1) + tanh b2) + tanh b3) + (((tanh b1) * tanh b2) * tanh b3)) * cosh b1) * cosh b2) * cosh b3 &
cosh ((b1 + b2) + b3) = (((((1 + ((tanh b1) * tanh b2)) + ((tanh b2) * tanh b3)) + ((tanh b3) * tanh b1)) * cosh b1) * cosh b2) * cosh b3 &
tanh ((b1 + b2) + b3) = ((((tanh b1) + tanh b2) + tanh b3) + (((tanh b1) * tanh b2) * tanh b3)) / (((1 + ((tanh b2) * tanh b3)) + ((tanh b3) * tanh b1)) + ((tanh b1) * tanh b2));
:: SIN_COS8:th 22
theorem
for b1, b2, b3 being real set holds
(((cosh (2 * b1)) + cosh (2 * b2)) + cosh (2 * b3)) + cosh (2 * ((b1 + b2) + b3)) = ((4 * cosh (b2 + b3)) * cosh (b3 + b1)) * cosh (b1 + b2);
:: SIN_COS8:th 23
theorem
for b1, b2, b3 being real set holds
(((((sinh b1) * sinh b2) * sinh (b2 - b1)) + (((sinh b2) * sinh b3) * sinh (b3 - b2))) + (((sinh b3) * sinh b1) * sinh (b1 - b3))) + (((sinh (b2 - b1)) * sinh (b3 - b2)) * sinh (b1 - b3)) = 0;
:: SIN_COS8:th 24
theorem
for b1 being real set
st 0 <= b1
holds sinh (b1 / 2) = sqrt (((cosh b1) - 1) / 2);
:: SIN_COS8:th 25
theorem
for b1 being real set
st b1 < 0
holds sinh (b1 / 2) = - sqrt (((cosh b1) - 1) / 2);
:: SIN_COS8:th 26
theorem
for b1 being real set holds
sinh (2 * b1) = (2 * sinh b1) * cosh b1 &
cosh (2 * b1) = (2 * ((cosh b1) ^2)) - 1 &
tanh (2 * b1) = (2 * tanh b1) / (1 + ((tanh b1) ^2));
:: SIN_COS8:th 27
theorem
for b1 being real set holds
sinh (2 * b1) = (2 * tanh b1) / (1 - ((tanh b1) ^2)) &
sinh (3 * b1) = (sinh b1) * ((4 * ((cosh b1) ^2)) - 1) &
sinh (3 * b1) = (3 * sinh b1) - ((2 * sinh b1) * (1 - cosh (2 * b1))) &
cosh (2 * b1) = 1 + (2 * ((sinh b1) ^2)) &
cosh (2 * b1) = (cosh b1) ^2 + ((sinh b1) ^2) &
cosh (2 * b1) = (1 + ((tanh b1) ^2)) / (1 - ((tanh b1) ^2)) &
cosh (3 * b1) = (cosh b1) * ((4 * ((sinh b1) ^2)) + 1) &
tanh (3 * b1) = ((3 * tanh b1) + ((tanh b1) |^ 3)) / (1 + (3 * ((tanh b1) ^2)));
:: SIN_COS8:th 28
theorem
for b1 being real set holds
(((sinh (5 * b1)) + (2 * sinh (3 * b1))) + sinh b1) / (((sinh (7 * b1)) + (2 * sinh (5 * b1))) + sinh (3 * b1)) = (sinh (3 * b1)) / sinh (5 * b1);
:: SIN_COS8:th 29
theorem
for b1 being real set
st 0 <= b1
holds tanh (b1 / 2) = sqrt (((cosh b1) - 1) / ((cosh b1) + 1));
:: SIN_COS8:th 30
theorem
for b1 being real set
st b1 < 0
holds tanh (b1 / 2) = - sqrt (((cosh b1) - 1) / ((cosh b1) + 1));
:: SIN_COS8:th 31
theorem
for b1 being real set holds
(sinh b1) |^ 3 = ((sinh (3 * b1)) - (3 * sinh b1)) / 4 &
(sinh b1) |^ 4 = (((cosh (4 * b1)) - (4 * cosh (2 * b1))) + 3) / 8 &
(sinh b1) |^ 5 = (((sinh (5 * b1)) - (5 * sinh (3 * b1))) + (10 * sinh b1)) / 16 &
(sinh b1) |^ 6 = ((((cosh (6 * b1)) - (6 * cosh (4 * b1))) + (15 * cosh (2 * b1))) - 10) / 32 &
(sinh b1) |^ 7 = ((((sinh (7 * b1)) - (7 * sinh (5 * b1))) + (21 * sinh (3 * b1))) - (35 * sinh b1)) / 64 &
(sinh b1) |^ 8 = (((((cosh (8 * b1)) - (8 * cosh (6 * b1))) + (28 * cosh (4 * b1))) - (56 * cosh (2 * b1))) + 35) / 128;
:: SIN_COS8:th 32
theorem
for b1 being real set holds
(cosh b1) |^ 3 = ((cosh (3 * b1)) + (3 * cosh b1)) / 4 &
(cosh b1) |^ 4 = (((cosh (4 * b1)) + (4 * cosh (2 * b1))) + 3) / 8 &
(cosh b1) |^ 5 = (((cosh (5 * b1)) + (5 * cosh (3 * b1))) + (10 * cosh b1)) / 16 &
(cosh b1) |^ 6 = ((((cosh (6 * b1)) + (6 * cosh (4 * b1))) + (15 * cosh (2 * b1))) + 10) / 32 &
(cosh b1) |^ 7 = ((((cosh (7 * b1)) + (7 * cosh (5 * b1))) + (21 * cosh (3 * b1))) + (35 * cosh b1)) / 64 &
(cosh b1) |^ 8 = (((((cosh (8 * b1)) + (8 * cosh (6 * b1))) + (28 * cosh (4 * b1))) + (56 * cosh (2 * b1))) + 35) / 128;
:: SIN_COS8:th 33
theorem
for b1, b2 being real set holds
(cosh (2 * b1)) + cos (2 * b2) = 2 + (2 * ((sinh b1) ^2 - ((sin b2) ^2))) &
(cosh (2 * b1)) - cos (2 * b2) = 2 * ((sinh b1) ^2 + ((sin b2) ^2));