Question

Prove the trigonometric identity. Show your work on the left hand side, or right hand side, or both.

Answer 1

$\bf sin^2(\theta)+cos^2(\theta)=1\implies sin^2(\theta)=1-cos^2(\theta)\\\\ \textit{also recall that }\qquad \textit{difference of squares} \\\\ (a-b)(a+b) = a^2-b^2\qquad \qquad a^2-b^2 = (a-b)(a+b)\\\\ -------------------------------\\\\ sin^4(x)-cos^4(x)=2sin^2(x)-1\\\\ -------------------------------\\\\ sin^4(x)-cos^4(x)\implies [sin^2(x)-cos^2(x)][sin^2(x)+cos^2(x)] \\\\\\\ [sin^2(x)-cos^2(x)][1]\implies sin^2(x)-cos^2(x) \\\\\\ sin^2(x)-[1-sin^2(x)]\implies sin^2(x)-1+sin^2(x)\implies 2sin^2(x)-1$