Question : If $\tan^4\theta + \tan^2\theta=1$, what is the value of $11(\cos^4\theta+\cos^2\theta)$?
Option 1: – 11
Option 2: 8
Option 3: 0
Option 4: 11
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Correct Answer: 11
Solution :
Given:
$\tan^4\theta + \tan^2\theta=1$
⇒ $\tan^2\theta(\tan^2\theta+1)=1 [\because \sec^2\theta-\tan^2\theta=1]$
⇒ $\tan^2\theta \sec^2\theta=1$
$\therefore \tan^2\theta=\cos^2\theta$
$11(\cos^4\theta+\cos^2\theta)$
= $11[(\cos^2\theta)^2+\cos^2\theta]$
= $11(\tan^4\theta + \tan^2\theta)$
= $11×1$
= $11$
Hence, the correct answer is 11.
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