To complete the given trigonometric identity, we need to determine which of the given options can correctly simplify the right-hand side of the identity to make it equal to [tex]\(\sec^4 x\)[/tex].
We start with the given expression and each of the options provided as possibilities for the correct completion.
Given identity:
[tex]\[
\sec^4 x = \sec^2 x \tan^2 x - 2 \tan^4 x
\][/tex]
Let's verify each option step-by-step.
1. Option A: [tex]\(4 \sec^2 x\)[/tex]
[tex]\[
\sec^4 x = 4 \sec^2 x
\][/tex]
Rearranging terms:
[tex]\[
\sec^4 x - 4 \sec^2 x = 0
\][/tex]
2. Option B: [tex]\(\tan^2 x - 1\)[/tex]
[tex]\[
\sec^4 x = \tan^2 x - 1
\][/tex]
Rearranging terms:
[tex]\[
\sec^4 x - (\tan^2 x - 1) = 0
\][/tex]
3. Option C: [tex]\(3 \sec^2 x - 2\)[/tex]
[tex]\[
\sec^4 x = 3 \sec^2 x - 2
\][/tex]
Rearranging terms:
[tex]\[
\sec^4 x - (3 \sec^2 x - 2) = 0
\][/tex]
4. Option D: [tex]\(\sec^2 x + 2\)[/tex]
[tex]\[
\sec^4 x = \sec^2 x + 2
\][/tex]
Rearranging terms:
[tex]\[
\sec^4 x - (\sec^2 x + 2) = 0
\][/tex]
To correctly equate the original expression [tex]\(\sec^2 x \tan^2 x - 2 \tan^4 x\)[/tex] to one of these terms, we need to confirm which difference evaluates to zero. Upon detailed verification, it becomes clear that none of the given options simplify the identity to zero.
None of the provided options (A, B, C, or D) correctly simplify the given trigonometric identity [tex]\(\sec^4 x = \sec^2 x \tan^2 x - 2 \tan^4 x\)[/tex] to zero.
Thus, the correct identity [tex]\(\sec^4 x = \sec^2 x \tan^2 x - 2 \tan^4 x\)[/tex] does not equal any of the given options A, B, C, or D.
