sin4θ – cos4θ = 1 – 2cos2θ

$\sin ^{4} \theta-\cos ^{4} \theta$

$=\left(\sin ^{2} \theta\right)^{2}-\left(\cos ^{2} \theta\right)^{2}$

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

$=1 \times\left(1-\cos ^{2} \theta-\cos ^{2} \theta\right) \quad\left(\sin ^{2} \theta+\cos ^{2} \theta=1\right)$

$=1-2 \cos ^{2} \theta$