Question : If $\tan \theta-\cot \theta=4$, then find the value of $\tan ^2 \theta+\cot ^2 \theta$.
Option 1: 16
Option 2: 20
Option 3: 14
Option 4: 18
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Correct Answer: 18
Solution : $\tan \theta-\cot \theta=4$ Squaring both sides, ⇒ $\tan^2\theta + \cot^2\theta - 2\tan\theta \cot\theta = 16$ ⇒ $\tan^2\theta + \cot^2\theta = 16+2$ $\therefore \tan^2\theta + \cot^2\theta = 18$ Hence, the correct answer is 18.
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Question : If $\theta$ be an acute angle and $\tan \theta+\cot \theta=2$, then the value of $2 \tan ^2 \theta+\cot ^2 \theta+\tan ^4 \theta \cot ^4 \theta$ is:
Option 1: 4
Option 2: 2
Option 3: 3
Option 4: 6
Question : If $\tan(5\theta-10^\circ)=\cot(5\varphi+20^\circ)$, then the value of $(\theta+\varphi)$ is:
Option 1: $16^\circ$
Option 2: $18^\circ$
Option 3: $15^\circ$
Option 4: $20^\circ$
Question : If $\tan \theta+\sin \theta=A$ and $\tan \theta-\sin \theta=B$, then what is the value of $A^2-B^2$?
Option 1: $\tan \theta \sin \theta$
Option 2: $4 \cot \theta$
Option 3: $4 \tan \theta \sin \theta$
Option 4: $2 \tan \theta \sin \theta$
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 $\sqrt{3} \tan ^2 \theta-4 \tan \theta+\sqrt{3}=0$, 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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