Question

The value of $\cos ({36}^o-A)\cos ({36}^o+A)+\cos ({54}^o+A)\cos ({54}^o-A)$ is?

• A. $\sin 2A$
• B. $\cos 2A$
• C. $\cos 3A$
• D. $\sin 3A$

Answer 1

The correct option is B $\cos 2A$

As we know that

$\cos (a+b)\cos (a-b)={\cos }^{2}a-{\sin }^{2}b$

$\cos ({36}^{\circ }-A)\cos ({36}^{\circ }+A)+\cos ({54}^{\circ }+A)\cos ({54}^{\circ }-A)={\cos }^2{36}^{\circ }-{\sin }^{2}A+{\cos }^2{54}^{\circ }-{\sin }^{2}A$

$={\cos }^2{36}^{\circ }+{\cos }^2({90}^{\circ }-{36}^{\circ })-2{\sin }^{2}A$

$={\cos }^2{36}^{\circ }+{\sin }^2{36}^{\circ }-2{\sin }^{2}A$

$=1-2{\sin }^{2}A=\cos 2A$