sin (a+b) = sinacosb + cosasinb
sin (a-b) = sinacosb - cosasinb
cos (a+b) = cosacosb - sinasinb
cos (a-b) = cosacosb + sinasinb
tan (a+b) = (tana+tanb)/(1-tanatanb)
tan (a-b) = (tana-tanb)/(1+tanatanb)

sin 2x = 2sinxcosx
cos 2x = (cosx)^(2) - (sinx)^(2)
tan 2x = (2tanx)/(1-(tanx)^(2))

cos x = √((1+cos(2x))/2)
sin x = √((1-cos(2x))/2)
tan x = √((1-cos(2x)/(1+cos(2x))

sinacosb = (1/2)(sin(a+b)+sin(a-b))
cosasinb = (1/2)(sin(a+b)-sin(a-b))
cosacosb = (1/2)(cos(a+b)+cos(a-b))
sinasinb = (-1/2)(cos(a+b)-cos(a-b))

sin a + sin b = 2sin([a+b]/2)cos([a-b]/2)
sin a - sin b = 2cos([a+b]/2)sin([a-b]/2)
cos a + cos b = 2cos([a+b]/2)cos([a-b]/2)
cos a - cos b = -2sin([a+b]/2)sin([a-b]/2)

dU = 𝛿q + 𝛿w ==> ΔU = q + w
𝛿w = -PdV ==> w = -PΔV

Isobaric Conditions ( ΔP = 0 )
𝛿q = nCp,m dT integrate both sides
q = nCp,mΔT ... (1)
𝛿w = -PdV, because P is constant, PdV = nRdT, therefore
𝛿w = -nRdT integrate both sides
w = -nRΔT = -PΔV ... (2)
from eq (1) and (2) we get ΔU = nΔT(Cp,m - R)