The value of $\begin{aligned} \cos \left(18^{\circ}-\mathrm{A}\right) \cdot \cos ( & \left…

The value of $\begin{aligned} \cos \left(18^{\circ}-\mathrm{A}\right) \cdot \cos ( & \left.18^{\circ}+\mathrm{A}\right) \\ & -\cos \left(72^{\circ}-\mathrm{A}\right) \cos \left(72^{\circ}+\mathrm{A}\right) \text { is }\end{aligned}$
  1. $\cos 72^{\circ}$
  2. $\sin 54^{\circ}$
  3. $\sin 18^{\circ}$
  4. $\cos 54^{\circ}$

Solution

$\begin{aligned} & \cos \left(18^{\circ}-\mathrm{A}\right) \cos \left(18^{\circ}+\mathrm{A}\right) \\ & \quad-\cos \left(72^{\circ}-\mathrm{A}\right) \cos \left(72^{\circ}+\mathrm{A}\right) \\ & =\cos \left(18^{\circ}-\mathrm{A}\right) \cos \left[90^{\circ}-\left(72^{\circ}-\mathrm{A}\right)\right] \\ & \quad-\cos \left(72^{\circ}-\mathrm{A}\right) \cos \left[90^{\circ}-\left(18^{\circ}-\mathrm{A}\right)\right] \\ & =\sin \left(72^{\circ}-\mathrm{A}\right) \cos \left(18^{\circ}-\mathrm{A}\right) \\ & \quad \quad-\cos \left(72^{\circ}-\mathrm{A}\right) \sin \left(18^{\circ}-\mathrm{A}\right) \\ & =\sin \left[\left(72^{\circ}-\mathrm{A}\right)-\left(18^{\circ}-\mathrm{A}\right)\right] \\ & =\sin 54^{\circ}\end{aligned}$

Asked in: MHT CET 2023 (13 May Shift 1)

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