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Prove. Prove that

can be written as

where ω_{0 }= 2π/T and

or as

where

Proof. The expression a_{n}cos nω_{0}t + b_{n}sin nω_{0}t 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.

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