Question : If $\theta$ is an acute angle and $\tan \theta+\cot \theta=2$, then the value of $\tan ^{200} \theta+\cot ^{200} \theta$ is:
Option 1: 1
Option 2: 2
Option 3: –1
Option 4: 0
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Correct Answer: 2
Solution :
Given $\theta$ is an acute angle
Also, $\tan \theta+\cot \theta=2$
$\frac{\sin\theta}{\cos\theta}+\frac{\cos\theta}{\sin\theta} = 2$
⇒ $\frac{\sin^2\theta + \cos^2\theta}{\sin\theta \cos\theta} = 2$
⇒ $1 = 2\sin\theta \cos\theta$
⇒ $\sin2\theta = 1$
⇒ $2\theta = 90°$
⇒ $\theta = 45°$
Now, $\tan ^{200} \theta+\cot ^{200} \theta$
$= (\tan 45°)^{200} + (\cot 45°)^{200} = 1^{200} + 1^{200} = 2$
Hence, the correct answer is 2.
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