Question : If $\sec^2 \theta+\tan^2 \theta=\frac{25}{18}$, the value of $\sec^4 \theta-\tan^4 \theta$ is:
Option 1: $\frac{18}{25}$
Option 2: $\frac{25}{12}$
Option 3: $\frac{25}{9}$
Option 4: $\frac{25}{18}$
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Correct Answer: $\frac{25}{18}$
Solution :
Given: $\sec^2 \theta+\tan^2 \theta=\frac{25}{18}$
To find: $\sec^4 \theta-\tan^4 \theta$
We know that $\sec^2 \theta-\tan^2 \theta=1$
Now, $\sec^4 \theta-\tan^4 \theta=(\sec^2 \theta+\tan^2 \theta)(\sec^2 \theta-\tan^2 \theta)$
$=(\frac{25}{18})(1)$
$=\frac{25}{18}$
Hence, the correct answer is $\frac{25}{18}$.
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