we are given
[tex] 1-2sin^2(\frac{x}{2} ) [/tex]
we can use trigonometric identity
we know that
[tex] sin^2(\frac{x}{2} )+cos^2(\frac{x}{2} )=1 [/tex]
now, we can replace 1 as this
[tex] =sin^2(\frac{x}{2} )+cos^2(\frac{x}{2} )-2sin^2(\frac{x}{2} ) [/tex]
now, we can simplify it
and we get
[tex] =cos^2(\frac{x}{2} )-sin^2(\frac{x}{2} ) [/tex]
now, we can use half angle formula
[tex] cos(\theta)=cos^2(\frac{\theta}{2} )-sin^2(\frac{\theta}{2} ) [/tex]
so, we will get
[tex] cos^2(\frac{x}{2} )-sin^2(\frac{x}{2} )=cos(2*\frac{x}{2}) [/tex]
[tex] =cos(x) [/tex]
so,
[tex] 1-2sin^2(\frac{x}{2} )=cos(x) [/tex].............Answer
