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JEE Advanced

Exam: JEE Advanced
Subject: Mathematics
Chapter: Trigonometry

JEE Advanced · Mathematics · Trigonometry

If ∑_ m=1 ^6 (α+(m-1) (π)/(4)) (α+(m π)/(4))=4 2 , where α ∈(0, π)...

If

\sum_{m=1}^6 \csc \left(\alpha+(m-1) \frac{\pi}{4}\right) \csc \left(\alpha+\frac{m \pi}{4}\right)=4 \sqrt{2}

, where

\alpha \in(0, \pi)

then

\alpha

can be:

Options

A $\frac{\pi}{12}$
B $\frac{\pi}{6}$
C $\frac{5 \pi}{12}$
D $\frac{\pi}{3}$

Correct answer

C. $\frac{5 \pi}{12}$

Step-by-step solution

\sum_{m=1}^6 \csc \left(\alpha+(m-1) \frac{\pi}{4}\right) \csc \left(\alpha+\frac{m \pi}{4}\right)=4 \sqrt{2}

On solving, we get

\sin 2 \alpha=\frac{1}{2}

Hence,

\quad 2 \alpha=\frac{\pi}{6}

or

\frac{5 \pi}{6} \quad \Rightarrow \quad \alpha=\frac{\pi}{12}

or

\frac{5 \pi}{12}

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