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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

NDA 2022 · Mathematics · Trigonometric Ratios & Identities

What is 36^ ^ - 72^ ^ equal to?

What is cos⁡36∘−cos⁡72∘\cos 36^{{}^{\circ}}-\cos 72^{{}^{\circ}} equal to?

Options

  1. A52\frac{\sqrt{5}}{2}
  2. B−52-\frac{\sqrt{5}}{2}
  3. C12\frac{1}{2}
  4. D−12-\frac{1}{2}

Correct answer

C. 12\frac{1}{2}

Step-by-step solution

cos⁡36∘−cos⁡72∘=−2sin⁡54∘⋅sin⁡(−18∘)=2sin⁡54∘⋅sin⁡18∘=2(5+14)(5−14)=12\begin{aligned} & \cos 36^{{}^{\circ}}-\cos 72^{{}^{\circ}}=-2 \sin 54^{{}^{\circ}} \cdot \sin \left(-18^{{}^{\circ}}\right) \\ & =2 \sin 54^{{}^{\circ}} \cdot \sin 18^{{}^{\circ}}=2\left(\frac{\sqrt{5}+1}{4}\right)\left(\frac{\sqrt{5}-1}{4}\right)=\frac{1}{2}\end{aligned}
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