Answer: sin(ωt)(1 - √2)
To simplify the expression cos(ωt+45°) + cos(ωt+135°) + cos(ωt-90°), we can make use of trigonometric identities.
Recall the following trigonometric identity:
cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
Using this identity, we can rewrite the expression as follows:
cos(ωt+45°) + cos(ωt+135°) + cos(ωt-90°)
= cos(ωt)cos(45°) - sin(ωt)sin(45°) + cos(ωt)cos(135°) - sin(ωt)sin(135°) + cos(ωt)cos(-90°) - sin(ωt)sin(-90°)
Now, let's simplify each term individually:
cos(45°) = √2/2
sin(45°) = √2/2
cos(135°) = -√2/2
sin(135°) = √2/2
cos(-90°) = 0
sin(-90°) = -1
Substituting these values back into the expression, we get:
cos(ωt)(√2/2) - sin(ωt)(√2/2) + cos(ωt)(-√2/2) - sin(ωt)(√2/2) + cos(ωt)(0) - sin(ωt)(-1)
Simplifying further:
(cos(ωt)√2/2 - cos(ωt)√2/2) + (-sin(ωt)√2/2 - sin(ωt)√2/2) - sin(ωt)(-1)
= 0 - 2sin(ωt)√2/2 + sin(ωt)
= -√2sin(ωt) + sin(ωt)
= sin(ωt)(1 - √2)
Therefore, the simplified expression is sin(ωt)(1 - √2).