Prove the trigonometric identity $\frac{\cos\left(x\right)^2-\sin\left(x\right)^2}{\cos\left(x\right)+\sin\left(x\right)}=\cos\left(x\right)-\sin\left(x\right)$

Starting from the left-hand side (LHS) of the identity

Simplify $\sqrt{\cos\left(x\right)^2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $\frac{1}{2}$

Any expression multiplied by $1$ is equal to itself

Simplify $\sqrt{\sin\left(x\right)^2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $\frac{1}{2}$

Any expression multiplied by $1$ is equal to itself

Simplify $\sqrt{\cos\left(x\right)^2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $\frac{1}{2}$

Simplify $\sqrt{\sin\left(x\right)^2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $\frac{1}{2}$

Simplify the fraction $\frac{\left(\cos\left(x\right)+\sin\left(x\right)\right)\left(\cos\left(x\right)-\sin\left(x\right)\right)}{\cos\left(x\right)+\sin\left(x\right)}$ by $\cos\left(x\right)+\sin\left(x\right)$

Since we have reached the expression of our goal, we have proven the identity