Final Answer
Step-by-step Solution
Specify the solving method
I. Express the LHS in terms of sine and cosine and simplify
Start from the LHS (left-hand side)
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 $\cos\left(x\right)+\sin\left(x\right)$
II. Express the RHS in terms of sine and cosine and simplify
Start from the RHS (right-hand side)
Nothing to do here. The expression is already in terms of sine and cosine and simplified
III. Choose what side of the identity are we going to work on
To prove an identity, we usually begin to work on the side of the equality that seems to be more complicated, or the side that is not expressed in terms of sine and cosine. In this problem, we will choose to work on the left side $\cos\left(x\right)-\sin\left(x\right)$ to reach the right side $\cos\left(x\right)-\sin\left(x\right)$
IV. Check if we arrived at the expression we wanted to prove
Since we have reached the expression of our goal, we have proven the identity