Question

Find the possible values of $\cos x\mathrm{if}cotx+\cos ecx=5$.

Answer 1

$$\begin{gathered} \mathrm{Given}:\cot x+\cos ecx=5 \\ \Rightarrow \frac{\cos x}{\sin x}+\frac{1}{\sin x}=5 \\ \Rightarrow \cos x+1=5\sin x \\ \mathrm{Squaring}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get}: \\ \Rightarrow {\left(\cos x+1\right)}^2={\left(5\sin x\right)}^2 \\ \Rightarrow {\cos }^{2}x+1+2\cos x=25{\sin }^{2}x \\ \Rightarrow {\cos }^{2}x+1+2\cos x=25(1-{\cos }^{2}x) \\ \Rightarrow 26{\cos }^{2}x+2\cos x-24=0 \\ \mathrm{Dividing}\mathrm{by}2\mathrm{and}\mathrm{splitting}\mathrm{the}\mathrm{middle}\mathrm{term},\mathrm{we}\mathrm{get}: \\ \Rightarrow 13{\cos }^{2}x+13\cos x-12\cos x-12=0 \\ \Rightarrow 13\cos x(\cos x+1)-12(\cos x+1)=0 \\ \Rightarrow \left(13\cos x-12\right)\left(\cos x+1\right)=0 \\ \mathrm{i}.\mathrm{e}.,\mathrm{either}\cos x=\frac{12}{13}\mathrm{or}\cos x=-1 \end{gathered}$$