Can someone check my work? j2ω / (1 + 2jω), if the unit is V(t) = cos t

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Problem : Use Fourier transform methods to find the time-domain
response network having a system function
j2ω / (1 + 2jω),
if the unit is
V(t) = cos t
(For a sinusoidal input cos t, the Fourier transform is
π[δ(ω−) + δ(ω − 1)]).


Solution : The time domain response for a particular input V(t) can be
Obtained by finding the product of the system function H(jω) and the
Fourier transform of the input. The inverse Fourier transform of the
resulting function is the time-domain response.
For a sinusoidal input cos t, the Fourier transform Pair
cos t <=> π[δ(ω + 1)] + δ(ω − 1)]
allows us to find the response
f(t) = F⁻¹ [{(j2ω) / (1 + 2j)} π{δ(ω + 1) + δ(ω −1)}]
f(t) = F⁻¹ [[{j2πω δ(ω + 1)} / (1 + 2jω)]
+ [{j2πω δ(ω − 1)} / (1 + 2jω)]]
f(t) = F⁻¹ [[{j2πω δ(ω + 1)} / {(1/2) + jω}]
+ [{jπω δ(ω − 1)} / {(1/2) + jω}]]
using the sifting property of the unit impulse, we obtain:
f(t) = F⁻¹ [−[{jπ δ(ω + 1)} / {(1/2) – j}] + [{jπ δ(ω − 1)} / {(1/2) + j}]]
f(t) = F⁻¹ [−[{jπ δ(ω + 1) ((1/2) + j)} / {(1/4) + 1}]
+ [{jπ δ(ω − 1) ((1/4) + 1)}]]
f(t) = F⁻¹ [[{π δ(ω + 1)} / (5/4)] − [{j(1/2)π δ(ω + 1)} / (5/4)]
+[{π δ(ω − 1)} / (5/4)] + [{j(1/2)π δ(ω − 1)} / (5/4)]]
f(t) = F⁻¹ [[(4/5)π{δ(ω + 1) + δ(ω − 1)}]
– [(2/5)π{j δ(ω + 1) − j δ(ω − 1)}]]
f(t) = F⁻¹ [(4/5)cos t – (2/5) sin t].


Thanks!