MHT CET · Maths · Trigonometric Ratios & Identities
The value of
\(\cos \left(18^{\circ}-\mathrm{A}\right) \cos \left(18^{\circ}+\mathrm{A}\right) \
-\cos \left(72^{\circ}-\mathrm{A}\right) \cos\) \(\left(72^{\circ}+\mathrm{A}\right)\) is equal to
- A \(\cos 54^{\circ}\)
- B \(\cos 36^{\circ}\)
- C \(\sin 54^{\circ}\)
- D \(\sin 36^{\circ}\)
Answer & Solution
Correct Answer
(C) \(\sin 54^{\circ}\)
Step-by-step Solution
Detailed explanation
\(\cos \left(18^{\circ}-\mathrm{A}\right) \cos \left(18^{\circ}+\mathrm{A}\right) \) \( -\cos \left(72^{\circ}-\mathrm{A}\right) \cos \left(72^{\circ}+\mathrm{A}\right) \)
\( =\cos \left(18^{\circ}-\mathrm{A}\right) \cos \left[90^{\circ}-\left(72^{\circ}-\mathrm{A}\right)\right] \) \( -\cos \left(72^{\circ}-\mathrm{A}\right) \cos \left[90^{\circ}-\left(18^{\circ}-\mathrm{A}\right)\right]\)
\(=\sin \left(72^{\circ}-\mathrm{A}\right) \cos \left(18^{\circ}-\mathrm{A}\right) \) \(-\cos \left(72^{\circ}-\mathrm{A}\right) \sin \left(18^{\circ}-\mathrm{A}\right) \)
\( =\sin \left[\left(72^{\circ}-\mathrm{A}\right)-\left(18^{\circ}-\mathrm{A}\right)\right]=\sin 54^{\circ}\)
\( =\cos \left(18^{\circ}-\mathrm{A}\right) \cos \left[90^{\circ}-\left(72^{\circ}-\mathrm{A}\right)\right] \) \( -\cos \left(72^{\circ}-\mathrm{A}\right) \cos \left[90^{\circ}-\left(18^{\circ}-\mathrm{A}\right)\right]\)
\(=\sin \left(72^{\circ}-\mathrm{A}\right) \cos \left(18^{\circ}-\mathrm{A}\right) \) \(-\cos \left(72^{\circ}-\mathrm{A}\right) \sin \left(18^{\circ}-\mathrm{A}\right) \)
\( =\sin \left[\left(72^{\circ}-\mathrm{A}\right)-\left(18^{\circ}-\mathrm{A}\right)\right]=\sin 54^{\circ}\)
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