Question

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

Answer 1

$\cot x+\csc x=5$

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

Squaring both sides we get,

$\Rightarrow 1+{\cos }^{2}x+2\cos x=5{\sin }^{2}x$

$\Rightarrow {\cos }^{2}x+2\cos x+1=5-5{\cos }^{2}x$

$\Rightarrow 6{\cos }^{2}x+2\cos x-4=0$

$\Rightarrow 6{\cos }^{2}x+6\cos x-4\cos x-4=0$

$\Rightarrow 6\cos x(\cos x+1)-4(\cos x+1)=0$

$\therefore \cos x=\frac{4}{6}or\cos x=-1$

$\Rightarrow \cos x=\frac{2}{3}or\cos x=-1$