How to Prove Sin(x/2) = +/- √((1 – cosx) / 2): A Step-by-Step Guide

sin(x/2) = +/- √((1 – cosx) / 2)

1

To prove that sin(x/2) = +/- √((1 – cosx) / 2), we can use the half-angle formula for sine and the Pythagorean identity.

First, let’s use the half-angle formula for sine:

sin(x/2) = ±√[(1 – cosx) / 2]

We need to prove that this equation is true for all values of x.

Start with the right-hand side of the equation:

±√[(1 – cosx) / 2]

We can simplify this expression by multiplying the numerator and denominator by 2:

±√[2(1 – cosx) / 2]

Simplifying further:

±√[1 – cosx]

Now, let’s use the half-angle formula for sine:

sin(x/2) = ±√[(1 – cosx) / 2] = ±√[1/2 – 1/2cosx] = ±√[1/2(1-cosx)]

Using the Pythagorean identity, we know that:

sin^2(x/2) + cos^2(x/2) = 1

Rearranging this equation, we get:

sin^2(x/2) = 1 – cos^2(x/2)

Taking the square root of both sides:

sin(x/2) = ±√[1 – cos^2(x/2)]

Now, we can substitute the half-angle formula for cosine into this equation:

sin(x/2) = ±√[1 – (1 + cosx)/2]
sin(x/2) = ±√[(2 – (1 + cosx)) / 2]
sin(x/2) = ±√[(1 – cosx) / 2]

This is the same as the right-hand side of the original equation, so we have shown that sin(x/2) = ±√[(1 – cosx) / 2] for all values of x.

More Answers: