Question

Find the possible value of $\cos x$, if $\cot x+\csc x=5$

Answer 1

$\frac{\cos x}{\sin x}+\frac{1}{\sin x}=5$

$\cos x+1=5\sin x$

${\cos }^{2}x+1+2\cos x=25{\sin }^{2}x$

${\cos }^{2}x+1+2\cos x=25(1-{\cos }^{2}x)$

${\cos }^{2}x+25{\cos }^{2}x+1+2\cos x-25=0$

$26{\cos }^{2}x+2\cos x-24=0$

$\cos x=\frac{-2\begin{array}{c}+\\ -\end{array}\sqrt{4-(4\times 26\times -24)}}{2\times 26}$

$\cos x=\frac{12}{13}or-1$