Answer:
(√2/2)(sin(x) +cos(x))
Explanation:
You want sin(x+π/4) written in terms of sine and cosine.
Angle sum identity
The sine of the sum of two angles is found using the identity ...
sin(α +β) = sin(α)cos(β) +cos(α)sin(β)
Application
Choosing α=x and β=π/4, this becomes ...
sin(x +π/4) = sin(x)cos(π/4) +cos(x)sin(π/4)
The trig function values are ...
sin(π/4) = cos(π/4) = √2/2, so the expression can be written ...
sin(x +π/4) = sin(x)√2/2 +cos(x)√2/2
sin(x +π/4) = (√2/2)(sin(x) +cos(x))
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