Proof 3 - 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 = r1(cos α + j sin α)

and

r2e = r2(cos β + j sin β)

We multiply these complex numbers together.

Multiplying the left hand sides:

r1e × r2e = r1r2ej(α+β)

We can write this answer as:

r1r2ej(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).

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