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

NDA 2022 · Mathematics · Trigonometric Ratios & Identities

What is value of ^2 15^ ^ + ^2 15^ ^ ?

What is value of cot⁡215∘+tan⁡215∘\cot ^2 15^{{}^{\circ}}+\tan ^2 15^{{}^{\circ}} ?

Options

  1. A12
  2. B14
  3. C838 \sqrt{3}
  4. D4

Correct answer

B. 14

Step-by-step solution

cot⁡215∘+tan⁡215∘=cosec⁡215∘−1+sec⁡215∘−1=cosec⁡215∘+sec⁡215∘−2\begin{aligned} & \cot ^2 15^{{}^{\circ}}+\tan ^2 15^{{}^{\circ}} \\ & =\operatorname{cosec}^2 15^{{}^{\circ}}-1+\sec ^2 15^{{}^{\circ}}-1 \\ & =\operatorname{cosec}^2 15^{{}^{\circ}}+\sec ^2 15^{{}^{\circ}}-2\end{aligned}=1sin⁡215∘+1cos⁡215∘−2=sin⁡215∘+cos⁡215∘14(4sin⁡215∘⋅cos⁡215∘)−2=4sin⁡230∘−2=414−2=16−2=14\begin{aligned} & =\frac{1}{\sin ^2 15^{{}^{\circ}}}+\frac{1}{\cos ^2 15^{{}^{\circ}}}-2 \\ & =\frac{\sin ^2 15^{{}^{\circ}}+\cos ^2 15^{{}^{\circ}}}{\frac{1}{4}\left(4 \sin ^2 15^{{}^{\circ}} \cdot \cos ^2 15^{{}^{\circ}}\right)}-2 \\ & =\frac{4}{\sin ^2 30^{{}^{\circ}}}-2=\frac{4}{\frac{1}{4}}-2=16-2=14\end{aligned}$
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