Question

The solution set of the trigonometric equation $\tan \theta+5 \cot \theta=\sec \theta$ is

• A. $\left\{\frac{\theta}{\theta}=2 n \pi \pm \frac{\pi}{3}, n \in Z\right\}$
• B. $\left\{\frac{\theta}{\theta}=n \pi+(-1)^{n} \frac{\pi}{2}, n \in Z\right\}$
• C. $\left\{\frac{\theta}{\theta}=n \pi+\frac{\pi}{6}, n \in Z\right\}$
• D. $\varphi$

Answer 1

Answer: (D)

We have
$\tan \theta+5 \cot \theta=\sec \theta$
$\Rightarrow \frac{\sin \theta}{\cos \theta}+\frac{5 \cos \theta}{\sin \theta}=\frac{1}{\cos \theta}, \sin \theta \neq 0, \cos \theta \neq 0$
$\Rightarrow \frac{\sin ^{2} \theta+5 \cos ^{2} \theta}{\cos \theta \sin \theta}=\frac{1}{\cos \theta}, \sin \theta \neq 0, \cos \theta \neq 0$
$\Rightarrow \sin ^{2} \theta+5\left(1-\sin ^{2} \theta\right)=\sin \theta, \sin \theta \neq 0, \cos \theta \neq 0$
$\Rightarrow 4 \sin ^{2} \theta+\sin \theta-5=0, \sin \theta \neq 0, \cos \theta \neq 0$
$\Rightarrow (4 \sin \theta+5)(\sin \theta-1)=0, \sin \theta \neq 0, \cos \theta \neq 0$
$\sin \theta=\frac{-5}{4}, 1, \sin \theta \neq 0, \cos \theta \neq 0$
$\Rightarrow \sin \theta=1, \sin \theta \neq 0, \cos \theta \neq 0[\because-1 \leq \sin \theta \leq 1]$
$\Rightarrow \theta \in \phi[$ if $\sin \theta=0 \Rightarrow \cos \theta=0]$