Question

Assertion :The value of $\theta ,0<\theta <\frac{\pi }{2},$ satisfying the equation $\sum _{m=1}^{6}cosec[\theta +\frac{(m-1)\pi }{4}]cosec[\theta +\frac{m\pi }{4}]=4\sqrt{2}$ are $\frac{\pi }{12}$ and $\frac{5\pi }{12}$ Reason: If $\tan \theta +\cot \theta =4,0<\theta <\frac{\pi }{2},$ then $\theta =\frac{\pi }{12}$ or $\frac{5\pi }{12}$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

In statement -1, L.H.S. is equal to
$\sum _{m=1}^6\frac{\sin [\theta +\frac{m\pi }{4}-(\theta +\frac{(m-1)\pi }{4})]}{\sin [\theta +\frac{(m-1)\pi }{4}]\sin [\theta +\frac{m\pi }{4}]\sin \frac{\pi }{4}}$
$\sqrt{2}\sum _{m=1}^6(\cos \left[\theta \frac{(m-1)\pi }{4}\right]-\cot [\theta +\frac{m\pi }{4}])$
$\sqrt{2}(\cot \theta -\cot \theta (\theta +\frac{\pi }{4})+\cot (\theta +\frac{\pi }{4})..........\cot (\theta +\frac{6\pi }{4}))$
$\sqrt{2}[\cot \theta -\cot (\theta +\frac{3\pi }{2})]=\sqrt{2}[\cot \theta +\tan \theta ]=4\sqrt{2}$ (given)
$\Rightarrow \cot \theta +\tan \theta =4\Rightarrow {\tan }^2\theta -4\tan \theta +1=0$
$\Rightarrow \tan \theta =2\pm \sqrt{3}$
We have $\tan ({45}^o\pm \tan {30}^o)=\frac{1\pm \frac{1}{\sqrt{3}}}{1\mp \frac{1}{\sqrt{3}}}=\frac{\sqrt{3}\pm 1}{\sqrt{3}\mp 1}=\frac{4\pm \sqrt{3}}{2}$
$=2\pm \sqrt{3}$
So from (1) $\tan \theta =\tan ({45}^o\pm {30}^o)$
$\Rightarrow \theta ={15}^o$ or ${75}^o$ or $\theta =\frac{\pi }{12}$ or $\frac{5\pi }{12}$
Showing that statement -1 and 2 both are true and statement 2 leads to statement 1.