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

NDA 2022 · Mathematics · Trigonometric Ratios & Identities

What is the value of ^2 165^ ^ + ^2 165^ ^ ?

What is the value of tan⁡2165∘+cot⁡2165∘\tan ^2 165^{{}^{\circ}}+\cot ^2 165^{{}^{\circ}} ?

Options

  1. A77
  2. B1414
  3. C434 \sqrt{3}
  4. D838 \sqrt{3}

Correct answer

B. 1414

Step-by-step solution

Concept: - tan⁡(180∘−θ)=−tan⁡θ\tan \left(180^{{}^{\circ}}-\theta\right)=-\tan \theta - cot⁡(180∘−θ)=−cot⁡θ\cot \left(180^{{}^{\circ}}-\theta\right)=-\cot \theta - tan⁡15∘=2−3\tan 15^{{}^{\circ}}=2-\sqrt{3} - cot⁡15∘=2+3\cot 15^{{}^{\circ}}=2+\sqrt{3} Formula used: (a+b)2=a2+b2+2ab(a−b)2=a2+b2−2ab\begin{aligned} & (a+b)^2=a^2+b^2+2 a b \\ & (a-b)^2=a^2+b^2-2 a b \end{aligned} Calculation:  Given, tan⁡2165∘+cot⁡2165∘⇒(tan⁡(180−15)∘)2+(cot⁡(180−15)∘)2⇒(−tan⁡215∘)+(−cot⁡215∘)⇒tan⁡215∘+cot⁡215∘⇒(2−3)2+(2+3)2⇒(4+3−43)+(4+3+43)⇒tan⁡2165∘+cot⁡2165∘=14\begin{aligned} & \text{ Given, } \tan ^2 165^{{}^{\circ}}+\cot ^2 165^{{}^{\circ}} \\ & \Rightarrow\left(\tan (180-15)^{{}^{\circ}}\right)^2+\left(\cot (180-15)^{{}^{\circ}}\right)^2 \\ & \Rightarrow\left(-\tan ^2 15^{{}^{\circ}}\right)+\left(-\cot ^2 15^{{}^{\circ}}\right) \\ & \Rightarrow \tan ^2 15^{{}^{\circ}}+\cot ^2 15^{{}^{\circ}} \\ & \Rightarrow(2-\sqrt{3})^2+(2+\sqrt{3})^2 \\ & \Rightarrow(4+3-4 \sqrt{3})+(4+3+4 \sqrt{3}) \\ & \Rightarrow \tan ^2 165^{{}^{\circ}}+\cot ^2 165^{{}^{\circ}}=14 \end{aligned}
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