7cosec theta -3 cot theta =7,then what is the value of 7cot theta-3 cosec theta  ?

Nov 14, 2016

${\left(7 \csc \theta - 3 \cot \theta\right)}^{2} - {\left(7 \cot \theta - 3 \csc \theta\right)}^{2}$

$= 49 \left({\csc}^{2} \theta - {\cot}^{2} \theta\right) - 9 \left({\csc}^{2} \theta - {\cot}^{2} \theta\right) - 42 \csc \theta \cot \theta + 42 \csc \theta \cot \theta$

$= 49 - 9 = 40$

So
${\left(7 \csc \theta - 3 \cot \theta\right)}^{2} - {\left(7 \cot \theta - 3 \csc \theta\right)}^{2} = 40$

$\implies {7}^{2} - {\left(7 \cot \theta - 3 \csc \theta\right)}^{2} = 40$

$\implies {\left(7 \cot \theta - 3 \csc \theta\right)}^{2} = 49 - 40 = 9$

$\implies 7 \cot \theta - 3 \csc \theta = \pm 3$

Nov 14, 2016

$\pm 3$

Explanation:

$\frac{7}{\sin} \theta - 3 \cos \frac{\theta}{\sin} \theta = 7$

$7 - 3 \cos \theta = 7 \sin \theta$

Put $X = \cos \theta$ and $Y = \sin \theta$ then ${X}^{2} + {Y}^{2} = 1$ and $7 - 3 X = 7 Y$

so ${X}^{2} + {\left(1 - \frac{3}{7} X\right)}^{2} = 1$

${X}^{2} + \cancel{1} + \frac{9}{49} {X}^{2} - \frac{6}{7} X = \cancel{1}$

$X = 0$ or $\frac{58}{7} X = 6 \implies X = \frac{21}{29}$

The solution $X = 0$ gives $\theta = \frac{\pi}{2}$ so

$7 \cot \left(\frac{\pi}{2}\right) - 3 \csc \left(\frac{\pi}{2}\right) = 0 - 3 = - 3$

For $\theta = \arccos \left(\frac{21}{29}\right)$ then $\cos \theta = \frac{21}{29}$, $\sin \theta = \frac{20}{29}$

so

$7 \cos \frac{\theta}{\sin} \theta - \frac{3}{\sin} \theta = \frac{\frac{7 \cdot 21}{29} - 3}{\frac{20}{29}} = \frac{147 - 87}{20} = 3$