The smallest positive value (in degrees) of $\theta$ for which $\tan \left(\theta+100^{\circ}\right)=\tan…

The smallest positive value (in degrees) of $\theta$ for which $\tan \left(\theta+100^{\circ}\right)=\tan \left(\theta+50^{\circ}\right) \tan (\theta) \tan \left(\theta-50^{\circ}\right)$ is valid, is
  1. $60^{\circ}$
  2. $45^{\circ}$
  3. $30^{\circ}$
  4. $15^{\circ}$

Solution

Given, $\tan \left(\theta+100^{\circ}\right)=\tan \left(\theta+50^{\circ}\right) \tan \theta \cdot \tan \left(\theta-50^{\circ}\right)$ $\Rightarrow \frac{\tan \left(\theta+100^{\circ}\right)}{\tan \left(\theta-50^{\circ}\right)}=\tan \left(\theta+50^{\circ}\right) \cdot \tan \theta$ $\begin{aligned} & \Rightarrow \frac{\sin \left(\theta+100^{\circ}\right) \cos \left(\theta-50^{\circ}\right)}{\cos \left(\theta+100^{\circ}\right) \cdot \sin \left(\theta-50^{\circ}\right)}=\frac{\sin \left(\theta+50^{\circ}\right) \cdot \sin \theta}{\cos \left(\theta+50^{\circ}\right) \cdot \cos \theta} \\ & \Rightarrow \frac{\sin \left(\theta+100^{\circ}\right) \cdot \cos \left(\theta-50^{\circ}\right)+\cos \left(\theta+100^{\circ}\right) \cdot \sin \left(\theta-50^{\circ}\right)}{\sin \left(\theta+100^{\circ}\right) \cdot \cos \left(\theta-50^{\circ}\right)-\cos \left(\theta+100^{\circ}\right) \cdot \sin \left(\theta-50^{\circ}\right)} \\ & =\frac{\sin \left(\theta+50^{\circ}\right) \cdot \sin \theta+\cos \left(\theta+50^{\circ}\right) \cdot \cos \theta}{\sin \left(\theta+50^{\circ}\right) \cdot \sin \theta-\cos \left(\theta+50^{\circ}\right) \cdot \cos \theta} \\ & \Rightarrow \frac{\sin \left(2 \theta+50^{\circ}\right)}{\sin \left(150^{\circ}\right)}=\frac{\cos \left(50^{\circ}\right)}{-\cos \left(2 \theta+50^{\circ}\right)} \\ & \Rightarrow \sin \left(2 \theta+50^{\circ}\right) \cdot \cos \left(2 \theta+50^{\circ}\right)+\cos \left(50^{\circ}\right) \cdot \sin \left(150^{\circ}\right)=0 \\ & \Rightarrow \sin \left(4 \theta+100^{\circ}\right)+\cos 50^{\circ}=0 \\ & \Rightarrow \sin \left(4 \theta+100^{\circ}\right)=-\cos \left(50^{\circ}\right)=\sin \left(220^{\circ}\right) \\ & \Rightarrow 4 \theta+100^{\circ}=220^{\circ} \Rightarrow 4 \theta=120^{\circ}\end{aligned}$ $\Rightarrow \theta=30^{\circ}$. (It is smallest positive value)

Asked in: AP EAMCET 2024 (19 May Shift 2)

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