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NDA 2020
Exam
NDA
Year
2020
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

NDA 2020 · Mathematics · Trigonometric Ratios & Identities

What is the value of 48^ ^ - 12^ ^ ?

What is the value of cos⁡48∘−cos⁡12∘\cos 48^{{}^{\circ}}-\cos 12^{{}^{\circ}} ?

Options

  1. A5−14\frac{\sqrt{5}-1}{4}
  2. B1−54\frac{1-\sqrt{5}}{4}
  3. C5+12\frac{\sqrt{5}+1}{2}
  4. D1−58\frac{1-\sqrt{5}}{8}

Correct answer

B. 1−54\frac{1-\sqrt{5}}{4}

Step-by-step solution

cos⁡48∘−cos⁡12∘=2⋅sin⁡(48∘+12∘2)⋅sin⁡(12∘−48∘2)=2⋅sin⁡30∘⋅sin⁡(−18∘)=−sin⁡18∘=−(5−14)=(1−54).\begin{aligned} & \cos 48^{{}^{\circ}}-\cos 12^{{}^{\circ}} \\ = & 2 \cdot \sin \left(\frac{48^{{}^{\circ}}+12^{{}^{\circ}}}{2}\right) \cdot \sin \left(\frac{12^{{}^{\circ}}-48^{{}^{\circ}}}{2}\right) \\ = & 2 \cdot \sin 30^{{}^{\circ}} \cdot \sin \left(-18^{{}^{\circ}}\right)=-\sin 18^{{}^{\circ}} \\ = & -\left(\frac{\sqrt{5}-1}{4}\right)=\left(\frac{1-\sqrt{5}}{4}\right).\end{aligned}
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