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
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Correct Answer: 4
Solution :
Given, $\tan \theta+\cot \theta=2$
We know, $\cot\theta=\frac{1}{\tan\theta}$
⇒ $\tan\theta+\frac{1}{\tan\theta}=2$
⇒ $\tan^2\theta+1=2\tan\theta$
⇒ $\tan^2\theta+1-2\tan\theta=0$
⇒ $(\tan\theta-1)^2=0$
⇒ $\tan\theta-1=0$
⇒ $\tan\theta = 1$ [As $\theta$ is an acute angle]
⇒ $\cot\theta=1$
Now consider, $2 \tan ^2 \theta+\cot ^2 \theta+\tan ^4 \theta \cot ^4 \theta$
= $2\times(1)^2+1^2+(1)^4\times(1)^4$
= $2+1+1=4$
Hence, the correct answer is 4.
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