91Ó°ÊÓ

The given expression is$\cos \left(\mathrm{Î\pm }-\mathrm{β}\right)\cos \left(\mathrm{Î\pm }+\mathrm{β}\right)={\cos }^2\mathrm{Î\pm }-{\sin }^2\mathrm{β}.$

We have to establish each 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,$$\begin{gathered} \cos \left(\mathrm{Î\pm }+\mathrm{β}\right)=\cos \mathrm{Î\pm }\cos \mathrm{β}-\sin \mathrm{Î\pm }\sin \mathrm{β} \\ \cos \left(\mathrm{Î\pm }-\mathrm{β}\right)=\cos \mathrm{Î\pm }\cos \mathrm{β}+\sin \mathrm{Î\pm }\sin \mathrm{β} \end{gathered}$$

L.H.S

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

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

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

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

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