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).