Question

if cot teta =9/12 then find the value of sinteta+costeta/ sinteta- costeta + secteta + cosecteta/ secteta -cosecteta​

Answer 1

$\mathfrak{\huge \underline{Question:}}$

$If ~cot \theta = \frac{9}{12}\\\\ then ~find ~the~ value ~of \\\\ \frac {sin \theta +cos \theta} {sin \theta- cos\theta} + \frac {sec \theta + cosec \theta} {sec \theta- cosec \theta}$

$\mathfrak{\huge \underline{Solution:}}$

$Cot \theta = \frac{Adjacent} {Perpendicular} =\frac {9}{12} = \frac{3}{4}\\\\ Hypotenuse^2= Base^2+Perpendicular^2 \\\\ H^2= 3^2+4^2 \\\\ H^2= 9+16= 25 \\\\ H=\sqrt{25 }= 5 \\\\ Now, \\\\ Cos \theta = \frac{Adjacent } {Hypotenuse}= \frac{3}{5} \\\\ sin \theta= \frac{Perpendicular} {Hypotenuse} = \frac{4}{5}$

Now, we know that :

$Sec \theta= \frac{1}{Cos \theta} = \frac{5} {3} \\\\ Cosec \theta = \frac {1}{sin \theta} = \frac{5}{4}$

Now, just put the values in

$= \frac {sin \theta +cos \theta} {sin \theta- cos\theta} + \frac {sec \theta + cosec \theta} {sec \theta- cosec \theta} \\\\ =\frac {\frac{4} {5} +\frac{3} {5} } {\frac{4} {5} - \frac{3} {5} } + \frac {\frac{5} {3} + \frac{5} {4} } {\frac{5} {3} - \frac{5} {4} }\\\\= \frac{\frac {7}{5}}{\frac{1}{5}}+\frac{\frac {20+15}{12}}{\frac{20-15}{12}} \\\\ =7+\frac{35}{5}= 7+7=\huge \boxed{14}$

Answer 2

$\mathfrak{\huge \underline{Question:}}$

$If ~cot \theta = \frac{9}{12}\\\\ then ~find ~the~ value ~of \\\\ \frac {sin \theta +cos \theta} {sin \theta- cos\theta} + \frac {sec \theta + cosec \theta} {sec \theta- cosec \theta}$

$\mathfrak{\huge \underline{Solution:}}$

$Cot \theta = \frac{Adjacent} {Perpendicular} =\frac {9}{12} = \frac{3}{4}\\\\ Hypotenuse^2= Base^2+Perpendicular^2 \\\\ H^2= 3^2+4^2 \\\\ H^2= 9+16= 25 \\\\ H=\sqrt{25 }= 5 \\\\ Now, \\\\ Cos \theta = \frac{Adjacent } {Hypotenuse}= \frac{3}{5} \\\\ sin \theta= \frac{Perpendicular} {Hypotenuse} = \frac{4}{5}$

Now, we know that :

$Sec \theta= \frac{1}{Cos \theta} = \frac{5} {3} \\\\ Cosec \theta = \frac {1}{sin \theta} = \frac{5}{4}$

Now, just put the values in

$= \frac {sin \theta +cos \theta} {sin \theta- cos\theta} + \frac {sec \theta + cosec \theta} {sec \theta- cosec \theta} \\\\ =\frac {\frac{4} {5} +\frac{3} {5} } {\frac{4} {5} - \frac{3} {5} } + \frac {\frac{5} {3} + \frac{5} {4} } {\frac{5} {3} - \frac{5} {4} }\\\\= \frac{\frac {7}{5}}{\frac{1}{5}}+\frac{\frac {20+15}{12}}{\frac{20-15}{12}} \\\\ =7+\frac{35}{5}= 7+7=\huge \boxed{14}$