Which Trigonometric Identity in being used to simplify this integral?

Elyse

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Hey, was wondering if anyone could tell me what trig identity was used to simply this integral?

. . . . .\(\displaystyle \displaystyle =\, \dfrac{1}{16}\, \int_0^{16}\, \left( \left[ \cos\left(\dfrac{3\pi}{8}t \right)\, +\, \cos\left(\dfrac{\pi}{4}t\right) \right]^2 \, +\, \left[\sin\left(\dfrac{3\pi}{8}t\right) \right]^2\right)\, dt\)

. . . . .\(\displaystyle \displaystyle =\, \dfrac{1}{16}\, \int_0^{16}\, \left[1\, +\, 2\,\cos\left(\dfrac{3\pi}{8}t\right)\, \cos\left(\dfrac{\pi}{4}t\right)\, +\, \left(\cos\left(\dfrac{\pi}{4}t\right)\right)^2 \right]\, dt\)

thank you :D:D
 

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Hey, was wondering if anyone could tell me what trig identity was used to simply this integral?

. . . . .\(\displaystyle \displaystyle =\, \dfrac{1}{16}\, \int_0^{16}\, \left( \left[ \cos\left(\dfrac{3\pi}{8}t \right)\, +\, \cos\left(\dfrac{\pi}{4}t\right) \right]^2 \, +\, \left[\sin\left(\dfrac{3\pi}{8}t\right) \right]^2\right)\, dt\)

. . . . .\(\displaystyle \displaystyle =\, \dfrac{1}{16}\, \int_0^{16}\, \left[1\, +\, 2\,\cos\left(\dfrac{3\pi}{8}t\right)\, \cos\left(\dfrac{\pi}{4}t\right)\, +\, \left(\cos\left(\dfrac{\pi}{4}t\right)\right)^2 \right]\, dt\)

thank you :D:D
Expand the first "squared" expression - what do you get?
 
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