Proof 2 - Using Complex Numbers

The exponential and polar forms of a complex number provide an easy way to prove the fundamental trigonometric identities.

Assume we have 2 complex numbers which we write as:

r₁e^jα = r₁(cos α + j sin α)

and

r₂e^jβ = r₂(cos β + j sin β)

We multiply these complex numbers together.

Multiplying the left hand sides:

r₁e^jα × r₂e^jβ = r₁r₂e^j(α+β)

We can write this answer as:

r₁r₂e^j(α+β) = r₁r₂(cos (α+β) + j sin (α+β)) ..... (1)

Multiplying the right hand sides:

r₁(cos α + j sin α) × r₂(cos β + j sin β)

= r₁ r₂(cos α cos β + j cos α sin β + j sin α cos β − sin α sin β)

= r₁ r₂(cos α cos β − sin α sin β + j (cos α sin β + sin α cos β)) .... (2)

[since j² = -1]

Now, equating (1) and (2) and dividing both parts by r₁ r₂:

cos (α+β) + j sin (α+β) = cos α cos β − sin α sin β + j (cos α sin β + sin α cos β)

Equating the real parts gives:

cos (α+β) = cos α cos β − sin α sin β

Equating the imaginary parts gives:

sin (α+β) = sin α cos β + cos α sin β

We would then proceed to replace β with (-β) as before, to obtain the identities for sin (α − β) and cos (α − β).