$\operatorname{Cos}\left(36^{\circ}-\mathrm{A}\right) \cos \left(36^{\circ}+\mathrm{A}\right)+\cos…

$\operatorname{Cos}\left(36^{\circ}-\mathrm{A}\right) \cos \left(36^{\circ}+\mathrm{A}\right)+\cos \left(54^{\circ}+\mathrm{A}\right) \cos \left(54^{\circ}-\mathrm{A}\right)=$
  1. $\operatorname{Cos} 2 \mathrm{~A}$
  2. $\operatorname{Cos} \mathrm{A}$
  3. $\operatorname{Sin} 2 \mathrm{~A}$
  4. $\operatorname{Sin} \mathrm{A}$

Solution

$\begin{array}{l} \cos \left(36^{\circ}-\mathrm{A}\right) \cos \left(36^{\circ}+\mathrm{A}\right)+\cos \left(54^{\circ}+\mathrm{A}\right) \cos \left(54^{\circ}-\mathrm{A}\right) \\ =\cos \left(36^{\circ}-\mathrm{A}\right) \cdot \cos \left(36^{\circ}+\mathrm{A}\right)+\cos \left[90^{\circ}-\left(36^{\circ}-\mathrm{A}\right)\right] \cdot \cos \left[90^{\circ}-\left(36^{\circ}+\mathrm{A}\right)\right] \end{array}$ $=\cos \left(36^{\circ}-\mathrm{A}\right) \cdot \cos \left(36^{\circ}+\mathrm{A}\right)+\sin \left(36^{\circ}-\mathrm{A}\right) \cdot \sin \left(36^{\circ}+\mathrm{A}\right)$ $=\cos \left[\left(36^{\circ}-\mathrm{A}\right)-\left(36^{\circ}+\mathrm{A}\right)\right]=\cos \left[36^{\circ}-\mathrm{A}-36^{\circ}-\mathrm{A}\right]=\cos (-2 \mathrm{~A})=\cos 2 \mathrm{~A}$

Asked in: MHT CET 2020 (16 Oct Shift 2)

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