Question

Verify sin^4x-cos^4x/sin^2x-cos^2x=1

Answer 1

`\bf sin^2(\theta)+cos^2(\theta)=1 \\\\\\ \textit{difference of squares} \\ \quad \\ (a-b)(a+b) = a^2-b^2\qquad \qquad a^2-b^2 = (a-b)(a+b)\\\\ -----------------------------\\\\ \cfrac{sin^4(x)-cos^4(x)}{sin^2(x)-cos^2(x)}=1\\\\ -----------------------------\\\\`


`\bf \cfrac{sin^4(x)-cos^4(x)}{sin^2(x)-cos^2(x)}\implies \cfrac{[sin^2(x)-cos^2(x)][sin^2(x)+cos^2(x)]}{sin^2(x)-cos^2(x)} \\\\\\ \cfrac{[sin^2(x)+cos^2(x)]}{1}\implies \cfrac{1}{1}\implies 1`