Question

$7cosec\varphi -3cot\varphi =7,$ Prove that $7cot\varphi -3cosec\varphi =3$

Answer 1

$7cosec\varphi -3\cot \varphi =7$

$\Rightarrow 7(cosec\varphi -1)=3\cot \varphi$

$\Rightarrow 49(cosec\varphi -1{)}^2=9{\cot }^2\varphi$ (squaring)

$\Rightarrow 49(cosec\varphi -1{)}^2=9(cose{c}^2\varphi -1)$

$\Rightarrow 49(cosec\varphi -1{)}^2=9(cosec\varphi -1\left)\right(cosec\varphi +1)$

$\Rightarrow 49(cosec\varphi -1)=9(cosec\varphi +1)$

$\Rightarrow 40cosec\varphi =58$

$\Rightarrow cosec\varphi =\frac{58}{40}$

$\Rightarrow \cot \varphi =\frac{7}{3}(cosec\varphi -1)=\frac{7}{3}(\frac{58}{40}-1)$

$=\frac{7}{3}\left(\frac{58-40}{40}\right)=\frac{7\left(6\right)}{40}=\frac{42}{40}$

$7\cot \varphi -3cosec\varphi$

$=7\left(\frac{42}{40}\right)-3\left(\frac{58}{40}\right)=\frac{294-174}{40}$

$=\frac{120}{40}$

$=3$.