Question

cosec theta = x+1/4x prove that cosec + cot =2x

Answer 1

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Answer 2

Answer:

$cosec \theta = x + \frac{1}{4x}$

we know

${cot}^{2} \theta = {cosec}^{2} \theta - 1 \\ \\ \: \: \: \: \: \: \: \: \: \: \: = ( {x + \frac{1}{4x} })^{2} - 1 \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = {x}^{2} + \frac{1}{16 {x}^{2} } + \frac{1}{2} - 1 \\ \\ = {x}^{2} + \frac{1}{16 {x}^{2} } - \frac{1}{2}$

$\therefore \: \: cot \theta = ± \sqrt{( {x- \frac{1}{4x} })^{2} } \\ \\ = ±(x - \frac{1}{4x} )$

Now

$cosec \theta + cot \theta \\ \\ = x + \frac{1}{4x} + x - \frac{1}{4x} \\ \\ = 2x$