Question : If $\sqrt{3} \tan ^2 \theta-4 \tan \theta+\sqrt{3}=0$, then what is the value of $\tan ^2 \theta+\cot ^2 \theta$?
Option 1: $\frac{4}{3}$
Option 2: $\frac{10}{3}$
Option 3: $3$
Option 4: $\frac{6}{5}$
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Correct Answer: $\frac{10}{3}$
Solution : $\sqrt{3} \tan ^2 \theta-4 \tan \theta+\sqrt{3}=0$ Multiplying by $\sqrt{3}$ on both sides, we get, $3\tan^2\theta−4\sqrt{3}\tan \theta+3=0$ ⇒ $3\tan^2\theta−3\sqrt{3}\tan\theta−\sqrt{3}\tan\theta+3=0$ ⇒ $3\tan\theta(\tan\theta−\sqrt{3})−\sqrt{3}(\tan\theta−\sqrt{3})=0$ ⇒ $(3\tan\theta−\sqrt{3})(\tan\theta−\sqrt{3})=0$ ⇒ $\tan\theta=\frac{1}{\sqrt{3}}$ or, $\tan\theta=\sqrt{3}$ ⇒ $\theta=30°$ or, $\theta=60°$ So, $\tan ^2 \theta+\cot ^2 \theta$ $=\tan^2 30° + \cot^2 30°$ $=\frac{1}{3}+3$ $=\frac{10}{3}$ Hence, the correct answer is $\frac{10}{3}$.
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Question : If $\sec \theta+\tan \theta=\frac{1}{\sqrt{3}}$, then the positive value of $\cot \theta+\cos \theta$ is:
Option 1: $\frac{3 \sqrt{3}}{2}$
Option 2: $\frac{\sqrt{3}}{2}$
Option 3: $\frac{2}{3 \sqrt{3}}$
Option 4: $\frac{2}{\sqrt{3}}$
Question : If $\tan \theta \cdot \tan 2 \theta=1$, then the value of $\cot 5 \theta$ is:
Option 1: $-1$
Option 2: $1$
Option 3: $-\sqrt{3}$
Option 4: $\sqrt{3}$
Question : If $\tan (90-\theta)=\frac{2}{\sqrt{3}}$, then the value of $2 \sqrt{3} \tan \theta+1$ is:
Option 1: 4
Option 2: 5
Option 3: 3
Option 4: 6
Question : If $\sqrt{2} \sec ^2 \theta-4 \sec \theta+2 \sqrt{2}=0$, then what is the value $\sin ^2 \theta+\tan ^2 \theta$?
Option 1: $\frac{1}{2}$
Option 2: $\frac{2}{3}$
Option 3: $\frac{5}{2}$
Option 4: $\frac{3}{2}$
Question : If $6 \sec \theta=10$, then find the value of $\frac{5 \operatorname{cosec} \theta-3 \cot \theta}{4 \cos \theta+3 \sin \theta}$.
Option 1: $\frac{2}{3}$
Option 2: $\frac{3}{2}$
Option 3: $\frac{5}{6}$
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