Question : Find the value of $\left(\tan ^2 \theta+\tan ^4 \theta\right)$.
Option 1: $\cot ^2 \theta-\tan ^2 \theta$
Option 2: $\ {\sec}^4 \theta-\ {\sec}^2 \theta$
Option 3: $\ {\sec}^4 \theta-\ {\sec}^4 \theta$
Option 4: $ \ {\sec}^4 \theta+\ {\sec}^2 \theta$
Correct Answer: $\ {\sec}^4 \theta-\ {\sec}^2 \theta$
Solution :
According to the question,
$\left(\tan ^2 \theta+\tan ^4 \theta\right)$
$=(\tan^{2}\theta)^{2} + \tan^{2}\theta$
$=(\sec^{2}\theta - 1)^{2} + \sec^{2}\theta - 1$
$=\sec^{4}\theta - 2 \sec^{2}\theta + 1 + \sec^{2}\theta - 1$
$=\sec^{4}\theta - \sec^{2}\theta$
Hence, the correct answer is $\sec^{4}\theta - \sec^{2}\theta$.
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