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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Trigonometric Equations

NDA 2023 · Mathematics · Trigonometric Equations

If (π θ)= (π θ), 0 θ (π)/(2) ; then what is the value of 8...

If tan⁡(πcos⁡θ)=cot⁡(πsin⁡θ),0<θ<π2\tan (\pi \cos \theta)=\cot (\pi \sin \theta), 0 \lt \theta \lt \frac{\pi}{2}; then what is the value of $8 \sin ^2\left(\theta+\frac{\pi}{4}\right)

Options

  1. A16
  2. B2
  3. C1
  4. D12\frac{1}{2}

Correct answer

C. 1

Step-by-step solution

 Given, tan⁡(πcos⁡θ)=cot⁡(πsin⁡θ) Let tan⁡(πcos⁡θ)=cot⁡(πsin⁡θ)=k⇒πcos⁡θ=tan⁡−1k,πsin⁡θ=cot⁡−1k Since, πcos⁡θ+πsin⁡θ=π2⇒cos⁡θ+sin⁡θ=12⇒sin⁡(θ+π4)=122⇒8sin⁡2(θ+π4)=1\begin{array}{ll}\text{ Given, } & \tan(\pi \cos \theta)=\cot (\pi \sin \theta) \\ \text{ Let } & \tan(\pi \cos \theta)=\cot (\pi \sin \theta)=\mathrm{k} \\ \Rightarrow & \pi \cos \theta=\tan ^{-1} \mathrm{k}, \pi \sin \theta=\cot ^{-1} \mathrm{k} \\ \text{ Since, } & \pi \cos \theta+\pi \sin \theta=\frac{\pi}{2} \\ \Rightarrow & \cos \theta+\sin \theta=\frac{1}{2} \\ \Rightarrow & \sin \left(\theta+\frac{\pi}{4}\right)=\frac{1}{2 \sqrt{2}} \\ \Rightarrow & 8 \sin ^2 \left(\theta+\frac{\pi}{4}\right)=1\end{array}
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