Question 187840
*[Tex \LARGE \frac{cos^4 (x)-sin^4(x)}{cos^2(x)- sin^2(x)}] ... Start with the given expression.



Let {{{y=cos(x)}}} and {{{z=sin(x)}}}



*[Tex \LARGE \frac{y^4-z^4}{y^2-z^2}] ... Replace each "cos" with "y". Replace each "sin" with "z".  



*[Tex \LARGE \frac{(y^2-z^2)(y^2+z^2)}{y^2-z^2}] Factor the numerator (using the difference of squares formula).



*[Tex \LARGE y^2+z^2] Cancel out the common terms and simplify.



*[Tex \LARGE cos^2(x)+sin^2(x)] Plug in {{{y=cos(x)}}} and {{{z=sin(x)}}}




*[Tex \LARGE 1] ... Use the identity *[Tex \LARGE cos^2(x)+sin^2(x)=1] to simplify.



So *[Tex \LARGE \frac{cos^4 (x)-sin^4(x)}{cos^2(x)- sin^2(x)}=1]