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Question

# If $\mathrm{cos}ecA+cotA=11/2$, then $\mathrm{tan}A=?$

A

$21/22$

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B

$15/16$

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C

$44/117$

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D

$117/43$

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Solution

## The correct option is C $44/117$The explanation for the correct option:Step 1. Using the identity $\mathbit{c}\mathbit{o}\mathbit{s}\mathbit{e}{\mathbit{c}}^{\mathbf{2}}\mathbit{A}\mathbf{}\mathbf{–}\mathbf{}\mathbit{c}\mathbit{o}{\mathbit{t}}^{\mathbf{2}}\mathbit{A}\mathbf{}\mathbf{=}\mathbf{}\mathbf{1},$Given, $\mathrm{cos}ecA+cotA=11/2\dots .\left(i\right)$$\begin{array}{rcl}& ⇒& \left(\mathrm{cos}ecA+cotA\right)\left(\mathrm{cos}ecA–cotA\right)=1\\ \left(11/2\right)\left(\mathrm{cos}ecA–cotA\right)& =& 1\\ \mathrm{cos}ecA–cotA& =& 2/11\dots .\left(ii\right)\end{array}$Step 2. Subtracting (ii) from (i),$\begin{array}{rcl}\mathrm{cos}ecA+cotA–\mathrm{cos}ecA+cotA& =& \left(11/2\right)–\left(2/11\right)\\ 2cotA& =& \frac{\left(121–4\right)}{22}\\ cotA& =& 117/44\\ \mathrm{tan}A& =& 1}{cotA}\\ & =& \mathbf{44}}{\mathbf{117}}\end{array}$Hence, The correct answer is option (C).

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