Trigonometric Equations

$ \begin{aligned} &\text { We have, } \tan \theta+5 \cot \theta=\sec \theta\\ &\begin{aligned} & \Rightarrow \frac{\sin \theta}{\cos \theta}+\frac{5 \cos \theta}{\sin \theta}=\frac{1}{\cos \theta^{\prime}}, \sin \theta \neq 0, \\ & \cos \theta \neq 0 \end{aligned} \end{aligned} $

$ \begin{aligned} & \Rightarrow \frac{\sin ^2 \theta+5 \cos ^2 \theta}{\cos \theta \sin \theta}=\frac{1}{\cos \theta}, \sin \theta \neq 0 \\ & \cos \theta \neq 0 \\ & \Rightarrow \sin ^2 \theta+5\left(1-\sin ^2 \theta\right)=\sin \theta, \sin \theta \neq 0 \\ & \cos \theta \neq 0 \\ & \Rightarrow 4 \sin ^2 \theta+\sin \theta-5=0, \sin \theta \neq 0 \\ & \cos \theta \neq 0 \\ & \Rightarrow(4 \sin \theta+5)(\sin \theta-1)=0, \sin \theta \neq 0 \\ & \cos \theta \neq 0 \\ & \sin \theta=\frac{-5}{4}, 1, \sin \theta \neq 0, \cos \theta \neq 0 \\ & \Rightarrow \sin \theta=1, \sin \theta \neq 0, \cos \theta \neq 0 \\ & \quad[\because-1 \leq \sin \theta \leq 1] \end{aligned} $

$ \begin{aligned} & \Rightarrow \theta \in \phi \\ & {[\text { if } \sin \theta=0 \Rightarrow \cos \theta=0]} \end{aligned} $