Question

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

• A. cos 2A
• B. sin 2A
• C. cos 2A
• D. 0

Answer 1

The correct option is A

cos 2A

$cos({36}^{\circ }-A)cos({36}^{\circ }+A)+cos({54}^{\circ }-A)cos({54}^{\circ }+A)cos[{90}^{\circ }-({54}^{\circ }+A\left)\right]cos\left[{90}^{\circ }\right({54}^{\circ }-A\left)\right]+cos({54}^{\circ }-A)cos({54}^{\circ }+A)=sin({54}^{\circ }+A)sin({54}^{\circ }-A)+cos({54}^{\circ }-A)cos({54}^{\circ }+A)[\because cos({90}^{\circ }-\theta )=sin\theta ]=cos({54}^{\circ }+A-{54}^{\circ }+A)\left[\begin{array}{c}\because cos(A-B)=cosA\\ cosB+sinAsinB\end{array}\right]$
= cos 2A