Second order Differential Equations Prove two basic trig ide

Second order Differential Equations

Prove two basic trig identities by using Euler’s formula to expand both sides of the identity e ^i(A+B) = e ^iA e ^iB.

Solution

we need to expand bith the sides of e^i(A+B) = e^iAe^iB

we know that,

e^ix = cosx + isinx

on left hand side we have e^i(A+B)

we can say that,

e^i(A+B) = cos(A+B) + isin(A+B)

we know that,

cos(A+B) = cosAcosB - sinAsinB

and sin(A+B) = sinAcosB + sinBcosA

so we can say that,

e^i(A+B) = cosAcosB - sinAsinB + isinAcosB + isinBcosA -----------------1)

now on right hand side we have,

e^iAe^iB

so we can say that

e^iAe^iB = (cosA+isinA)(cosB+isinB)

e^iAe^iB = cosAcosB + isinBcosA + isinAcosB + i²sinAsinB

e^iAe^iB = cosAcosB + isinBcosA + isinAcosB - sinAsinB

e^iAe^iB = cosAcosB - sinAsinB + isinAcosB + isinBcosA --------------2)

so from equation 1) and 2) we can say that,

e^i(A+B) = e^iAe^iB