Question

Find the possible value of cos x , if cot x +cosec x= 5.

Answer 1

$\begin{array}{c}\cot x+\cos ecx=5\\ \\ \frac{\cos x}{\sin x}+\frac{1}{\sin x}=5\\ \\ \cos x+1=5sinx\end{array}$

squaring both sides

$\begin{array}{c}{\left(\cos x+1\right)}^2={\left(5\sin x\right)}^2\\ {\cos }^{2}x+1+2\cos x=25{\sin }^{2}x\\ {\cos }^{2}x+1+2\cos x=25\left(1-{\cos }^{2}x\right)\\ 26{\cos }^{2}x+2\cos x-24=0\end{array}$

dividing by $2$ we get

$\begin{array}{c}13{\cos }^{2}x+13\cos x-12\cos x-12=0\\ 13\cos x\left(\cos x+1\right)-12\left(\cos x+1\right)=0\\ \left(13\cos x-12\right)\left(\cos x+1\right)\\ \cos x=\frac{12}{13}\\ or\\ \cos x=-1\end{array}$