MHT CET · Maths · Trigonometric Ratios & Identities
\(\operatorname{Cos}\left(36^{\circ}-\mathrm{A}\right) \cos \left(36^{\circ}+\mathrm{A}\right)+\cos \left(54^{\circ}+\mathrm{A}\right) \cos \left(54^{\circ}-\mathrm{A}\right)=\)
- A \(\operatorname{Cos} 2 \mathrm{~A}\)
- B \(\operatorname{Cos} \mathrm{A}\)
- C \(\operatorname{Sin} 2 \mathrm{~A}\)
- D \(\operatorname{Sin} \mathrm{A}\)
Answer & Solution
Correct Answer
(A) \(\operatorname{Cos} 2 \mathrm{~A}\)
Step-by-step Solution
Detailed explanation
\(\cos \left(36^{\circ}-\mathrm{A}\right) \cos \left(36^{\circ}+\mathrm{A}\right)+\cos \left(54^{\circ}+\mathrm{A}\right) \cos \left(54^{\circ}-\mathrm{A}\right) \)
\( =\cos \left(36^{\circ}-\mathrm{A}\right) \cdot \cos \left(36^{\circ}+\mathrm{A}\right)+\cos [90^{\circ}-(36^{\circ}-\) \( \mathrm{A})] \cdot \cos \left[90^{\circ}-\left(36^{\circ}+\mathrm{A}\right)\right]\)
\(=\cos \left(36^{\circ}-\mathrm{A}\right) \cdot \cos \left(36^{\circ}+\mathrm{A}\right)+\sin \left(36^{\circ}-\mathrm{A}\right) \cdot \sin \left(36^{\circ}+\mathrm{A}\right)\)
\(=\cos \left[\left(36^{\circ}-\mathrm{A}\right)-\left(36^{\circ}+\mathrm{A}\right)\right]=\cos \left[36^{\circ}-\mathrm{A}-36^{\circ}-\mathrm{A}\right]=\cos (-2 \mathrm{~A})=\cos 2 \mathrm{~A}\)
\( =\cos \left(36^{\circ}-\mathrm{A}\right) \cdot \cos \left(36^{\circ}+\mathrm{A}\right)+\cos [90^{\circ}-(36^{\circ}-\) \( \mathrm{A})] \cdot \cos \left[90^{\circ}-\left(36^{\circ}+\mathrm{A}\right)\right]\)
\(=\cos \left(36^{\circ}-\mathrm{A}\right) \cdot \cos \left(36^{\circ}+\mathrm{A}\right)+\sin \left(36^{\circ}-\mathrm{A}\right) \cdot \sin \left(36^{\circ}+\mathrm{A}\right)\)
\(=\cos \left[\left(36^{\circ}-\mathrm{A}\right)-\left(36^{\circ}+\mathrm{A}\right)\right]=\cos \left[36^{\circ}-\mathrm{A}-36^{\circ}-\mathrm{A}\right]=\cos (-2 \mathrm{~A})=\cos 2 \mathrm{~A}\)
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