Question

In Problems 47–72, establish each identity.

$\cos \left(\alpha -\beta \right)\cos \left(\alpha +\beta \right)={\cos }^2\alpha -{\sin }^2\beta$

Short answer

The given identity is established.

Step-by-step solution

Step 1. Given Information 

The given expression is$\cos \left(\alpha -\beta \right)\cos \left(\alpha +\beta \right)={\cos }^2\alpha -{\sin }^2\beta .$

We have to establish each identity.

Step 2. Establishing the identity 

We can establish the given expression as an identity by simplifying one side to give the other side.

So, use the sum and difference identity for the cos function,$$\begin{gathered} \cos \left(\alpha +\beta \right)=\cos \alpha \cos \beta -\sin \alpha \sin \beta \\ \cos \left(\alpha -\beta \right)=\cos \alpha \cos \beta +\sin \alpha \sin \beta \end{gathered}$$

L.H.S

role="math" localid="1646477860296" $$\begin{gathered} \cos \left(\alpha -\beta \right)\cos \left(\alpha +\beta \right) \\ =\left(\cos \alpha \cos \beta +\sin \alpha \sin \beta \right)\left(\cos \alpha \cos \beta -\sin \alpha \sin \beta \right) \\ ={\left(\cos \alpha \cos \beta \right)}^2-{\left(\sin \alpha \sin \beta \right)}^2...........\left(i\right) \end{gathered}$$

As we know $$\begin{gathered} {\sin }^2\beta +{\cos }^2\beta =1 \\ {\cos }^2\beta =1-{\sin }^2\beta \end{gathered}$$

Now, substitute ${\cos }^2\beta$in equation (i)

role="math" localid="1646478364838" $$\begin{gathered} \cos \left(\alpha -\beta \right)\cos \left(\alpha +\beta \right) \\ ={\left(\cos \alpha \cos \beta \right)}^2-{\left(\sin \alpha \sin \beta \right)}^2 \\ ={\cos }^2\alpha \left(1-{\sin }^2\beta \right)-{\sin }^2\alpha {\sin }^2\beta \\ ={\cos }^2\alpha -{\cos }^2\alpha {\sin }^2\beta -{\sin }^2\alpha {\sin }^2\beta \\ ={\cos }^2\alpha -{\sin }^2\beta \left({\cos }^2\alpha +{\sin }^2\alpha \right)\left[{\cos }^2\alpha +{\sin }^2\alpha =1\right] \\ ={\cos }^2\alpha -{\sin }^2\beta \end{gathered}$$

Thus, L.H.S = R.H.S and the given expression is an identity.