What is 1-cosx Equal To?
Answer: 1 – cos(x) is equal to 2 sin²(x/2).
To derive this identity, let’s use the double-angle formula for sine:
sin(2θ) = 2sin(θ)cos(θ).
Now, set 2θ = x:
sin(x) = 2sin(x/2)cos(x/2).
Next, isolate cos(x/2):
cos(x/2) = (sin(x))/(2sin(x/2)).
Substitute this into 1 – cos(x):
1 – cos(x) = 1 – (sin(x))/(2sin(x/2)).
To rationalize the denominator, multiply both numerator and denominator by 2sin(x/2):
1 – cos(x) = (2sin(x/2) – sin(x))/(2sin(x/2)).
Now, factor out a 2sin(x/2) from the numerator:
1 – cos(x) = (2sin(x/2)(1 – sin(x/2)))/(2sin(x/2)).
Cancel out the common factor of 2sin(x/2):
1 – cos(x) = 1 – sin(x/2).
So, 1 – cos(x) simplifies to 1 – sin(x/2), which is also equal to 2 sin²(x/2).
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