Question : If $\tan\theta +\cot\theta =2$, then the value of $\left (\tan^{n}\theta+\cot^{n}\theta \right)$ is:
Option 1: $2^{n}$
Option 2: $2^{\frac{n}{2}}$
Option 3: $2^{\frac{1}{2}}$
Option 4: $2$
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Correct Answer: $2$
Solution : Given: $\tan\theta +\cot\theta =2$ Putting $\theta = 45^\circ$, LHS $=\tan 45^\circ + \cot 45^\circ = 1 + 1 =2=$ RHS (satisfies) So, $\tan^{n}\theta+\cot^{n}\theta$ $=\tan^{n}45^\circ+\cot^{n}45^\circ$ $= 1 + 1$ $= 2$ Hence, the correct answer is $2$.
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