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NDA 2024 Maths · Mathematics · Trigonometric Functions and Identities · medium

In surd form, cos 105^° equals

Single correct option

In surd form, cos⁡105∘\cos105^\circ equals
  1. A. 6−24\dfrac{\sqrt6-\sqrt2}{4}
  2. B. 6+24\dfrac{\sqrt6+\sqrt2}{4}
  3. C. 2−64\dfrac{\sqrt2-\sqrt6}{4}
  4. D. −32-\dfrac{\sqrt3}{2}
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Answer: (C) 2−64\dfrac{\sqrt2-\sqrt6}{4}

cos⁡(60∘+45∘)=12⋅12−32⋅12=1−322=2−64\cos(60^\circ+45^\circ)=\frac12\cdot\frac1{\sqrt2}-\frac{\sqrt3}2\cdot\frac1{\sqrt2}=\frac{1-\sqrt3}{2\sqrt2}=\frac{\sqrt2-\sqrt6}4, which is negative as 105∘105^\circ is in the second quadrant. Trap: dropping the sign.
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