sec4θ – tan4θ = 1 + 2 tan2θ

$\sec ^{4} \theta-\tan ^{4} \theta$

$=\left(\sec ^{2} \theta\right)^{2}-\left(\tan ^{2} \theta\right)^{2}$

$=\left(\sec ^{2} \theta-\tan ^{2} \theta\right)\left(\sec ^{2} \theta+\tan ^{2} \theta\right) \quad\left[a^{2}-b^{2}=(a-b)(a+b)\right]$

$=1 \times\left(1+\tan ^{2} \theta+\tan ^{2} \theta\right) \quad\left(1+\tan ^{2} \theta=\sec ^{2} \theta\right)$

$=1+2 \tan ^{2} \theta$