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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

NDA 2024 · Mathematics · Trigonometric Ratios & Identities

Consider the following numbers: A. 22.5^ ^ B. 22.5^ ^ C. 22.5^ ^ -...

Consider the following numbers: A. tan⁡22.5∘\tan 22.5^{{}^{\circ}} B. cot⁡22.5∘\cot 22.5^{{}^{\circ}} C. tan⁡22.5∘−cot⁡22.5∘\tan 22.5^{{}^{\circ}}-\cot 22.5^{{}^{\circ}} How many of the above are irrational numbers?

Options

  1. ANone
  2. BOnly one
  3. COnly two
  4. DAll three

Correct answer

B. Only one

Step-by-step solution

Let t=tan⁡22.5∘=tan⁡45∘2\mathrm{t}=\tan 22.5^{{}^{\circ}}=\tan \frac{45^{{}^{\circ}}}{2} Now, tan⁡45∘=tan⁡(2×452)∘=2tan⁡(452)∘1−tan⁡2(452)∘\tan 45^{{}^{\circ}}=\tan \left(2 \times \frac{45}{2}\right)^{{}^{\circ}}=\frac{2 \tan \left(\frac{45}{2}\right)^{{}^{\circ}}}{1-\tan ^2\left(\frac{45}{2}\right)^{{}^{\circ}}}⇒1=2t1−t2⇒t2+2t−1=0⇒t=−2±4+42=−1±2⇒tan⁡(452)∘=−1±2⇒tan⁡(452)∘=−1+(2)\begin{aligned} & \Rightarrow 1=\frac{2 t}{1-t^2} \Rightarrow t^2+2 t-1=0 \\ & \Rightarrow t=\frac{-2 \pm \sqrt{4+4}}{2}=-1 \pm \sqrt{2} \\ & \Rightarrow \tan \left(\frac{45}{2}\right)^{{}^{\circ}}=-1 \pm \sqrt{2} \Rightarrow \tan \left(\frac{45}{2}\right)^{{}^{\circ}}=-1+(\sqrt{2}) \end{aligned}{∴ In \left\{\therefore\right.\text{ In }\mathrm{I}st  quadrant tan⁡ is  positive }^{\text{st }}\text{ quadrant } \tan\text{ is }\text{ positive } \} (II) Now, cot⁡22.5∘=1tan⁡22.5∘=1−1+2=1+2\cot 22.5^{{}^{\circ}}=\frac{1}{\tan 22.5^{{}^{\circ}}}=\frac{1}{-1+\sqrt{2}}=1+\sqrt{2} which is also an irrational number. (III) Since, tan⁡22.5∘−cot⁡22.5∘=−1+2−(1+2)=−2\tan 22.5^{{}^{\circ}}-\cot 22.5^{{}^{\circ}}=-1+\sqrt{2}-(1+\sqrt{2})=-2 which is not an irrational number.
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