Question

Evaluate: ${\left(a-b\right)}^2{\cos }^2\frac{c}{2}+{\left(a+b\right)}^2{\sin }^2\frac{c}{2}=\left(a+b\right)\left[a+b\left({\sin }^2\frac{c}{2}-{\cos }^2\frac{c}{2}\right)\right]$

Answer 1

Given, ${\left(a-b\right)}^2{\cos }^2\frac{c}{2}+{\left(a+b\right)}^2{\sin }^2\frac{c}{2}$

$=\left(a-b\right)\left(a+b\right){\cos }^2\frac{c}{2}+\left(a+b\right)\left(a+b\right){\sin }^2\frac{c}{2}$

$=\left(a+b\right)\left[\left(a-b\right){\cos }^2\frac{c}{2}+\left(a+b\right){\sin }^2\frac{c}{2}\right]$

$=\left(a+b\right)\left[a{\cos }^2\frac{c}{2}-b{\cos }^2\frac{c}{2}+a{\sin }^2\frac{c}{2}+b{\sin }^2\frac{c}{2}\right]$

$=\left(a+b\right)\left[a{\cos }^2\frac{c}{2}+a{\sin }^2\frac{c}{2}+b{\sin }^2\frac{c}{2}-b{\cos }^2\frac{c}{2}\right]$

$=\left(a+b\right)\left[a\left({\sin }^2\frac{c}{2}+{\cos }^2\frac{c}{2}\right)+b{\sin }^2\frac{c}{2}-b{\cos }^2\frac{c}{2}\right]$

$=\left(a+b\right)\left[a+b\left({\sin }^2\frac{c}{2}-{\cos }^2\frac{c}{2}\right)\right]$