Question : If $\cos^2x+\cos^4x=1$, then $\tan^2x+\tan^4x$?
Option 1: $0$
Option 2: $1$
Option 3: $2 \tan^2x$
Option 4: $2\tan^4x$
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Correct Answer: $1$
Solution :
$\cos^2x+\cos^4x=1$
⇒ $\cos^4x = 1 - \cos^2x$
⇒ $\cos^4x = \sin^2x$ [Using $1-\cos^2x=\sin^2x$]
⇒ $\cos^2x\times \cos^2x=\sin^2x$
⇒ $\tan^2x=\cos^2x$ --------(i)
On squaring,
$\tan^4x=\cos^4x$ ------(ii)
⇒ $\tan^2x+\tan^4x = \cos^2x+\cos^4x$
⇒ $\tan^2x+\tan^4x=1$
Hence, the correct answer is $1$.
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