Question : If $\tan ^2 \theta+\tan ^4 \theta=1$, then:
Option 1: $\cot ^2 \theta+\cot ^4 \theta=1$
Option 2: $\cos ^2 \theta+\cos ^4 \theta=1$
Option 3: $\sin ^2 \theta+\sin ^4 \theta=1$
Option 4: $\operatorname{cosec}^2 \theta+\sec ^4 \theta=1$
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Correct Answer: $\cos ^2 \theta+\cos ^4 \theta=1$
Solution :
$\tan ^2 \theta+\tan ^4 \theta=1$
⇒ $\tan^2\theta(1+\tan^2\theta) = 1$
⇒ $\tan^2\theta(1+\frac{\sin^2\theta}{\cos^2\theta}) = 1$
⇒ $\frac{\sin^2\theta}{\cos^2\theta}(\frac{cos^2\theta + \sin^2\theta}{\cos^2\theta}) = 1$
⇒ $\frac{\sin^2\theta}{\cos^2\theta}(\frac{1}{\cos^2\theta}) = 1$
⇒ $\sin^2\theta = \cos^4\theta$
⇒ $1-\cos^2\theta = \cos^4\theta$
⇒ $\cos^4\theta + \cos^2\theta = 1$
Hence, the correct answer is $\cos^2\theta + \cos^4\theta = 1$.
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