Question

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

Answer 1

Determine the value of $\cos \left({36}^{\mathrm{o}}-\mathrm{A}\right)\cos \left({36}^{\mathrm{o}}+\mathrm{A}\right)+\cos \left({54}^{\mathrm{o}}-\mathrm{A}\right)\cos \left({54}^{\mathrm{o}}+\mathrm{A}\right)$.

Since,$\cos ({54}^{\mathrm{o}}+\mathrm{A})=\sin [{90}^{\mathrm{o}}-({54}^{\mathrm{o}}+\mathrm{A}\left)\right]$ and $\cos ({54}^{\mathrm{o}}-\mathrm{A})=\sin [{90}^{\mathrm{o}}-({54}^{\mathrm{o}}+\mathrm{A}\left)\right]$ [using $\cos \mathrm{\theta }=\sin ({90}^{\mathrm{o}}-\mathrm{\theta })$]

Now substitute the above value in the given expression:

$\begin{array}{rcl}\cos \left({36}^{\mathrm{o}}-\mathrm{A}\right)\cos \left({36}^{\mathrm{o}}+\mathrm{A}\right)+\cos \left({54}^{\mathrm{o}}-\mathrm{A}\right)\cos \left({54}^{\mathrm{o}}+\mathrm{A}\right)& =& \cos \left({36}^{\mathrm{o}}-\mathrm{A}\right)\cos \left({36}^{\mathrm{o}}+\mathrm{A}\right)+\sin \left[{90}^{\mathrm{o}}-\left({54}^{\mathrm{o}}-\mathrm{A}\right)\right]\sin \left[{90}^{\mathrm{o}}-\left({54}^{\mathrm{o}}+\mathrm{A}\right)\right]\\ & =& \cos \left({36}^{\mathrm{o}}-\mathrm{A}\right)\cos \left({36}^{\mathrm{o}}+\mathrm{A}\right)+\sin \left({36}^{\mathrm{o}}+\mathrm{A}\right)\sin \left({36}^{\mathrm{o}}-\mathrm{A}\right)\left[\because \cos (\mathrm{A}-\mathrm{B})=\cos \mathrm{A}\cos \mathrm{B}+\sin \mathrm{A}\sin \mathrm{B}\right]\\ & =& \cos \left({36}^{\mathrm{o}}+\mathrm{A}\right)\cos \left({36}^{\mathrm{o}}-\mathrm{A}\right)+\sin \left({36}^{\mathrm{o}}+\mathrm{A}\right)\sin \left({36}^{\mathrm{o}}-\mathrm{A}\right)\\ & =& \cos [\left({36}^{\mathrm{o}}+\mathrm{A}\right)-\left({36}^{\mathrm{o}}-\mathrm{A}\right)]\\ & =& \cos \left(2\mathrm{A}\right)\end{array}$Hence, the required value of $\cos \left({36}^{\mathrm{o}}-\mathrm{A}\right)\cos \left({36}^{\mathrm{o}}+\mathrm{A}\right)+\cos \left({54}^{\mathrm{o}}-\mathrm{A}\right)\cos \left({54}^{\mathrm{o}}+\mathrm{A}\right)=\cos \left(2\mathrm{A}\right)$