Prove. Prove that

can be written as

where ω₀ = 2π/T and

or as

where

Proof. The expression a_ncos nω₀t + b_nsin nω₀t in 1) can be written as

Let us now define the angles θ_n and in accordance with the triangle shown in Fig. 1 and set

With this notation 4) can be written as

Using a formula from trigonometry that states that

            cos (A - B) = cos A cos B + sin A sin B

we can write 5) as  

Using this result, 1) can be written as

which is the result required for 2) above. To obtain result 3) we note that 4) can also be written as  

Using a formula from trigonometry that states that

            sin (A + B) = sin A cos B + sin B cos A

we can write 8) as  

Thus 1) can be written as

which is the result required for 3) above.