Question 190819
*[Tex \LARGE \sec(x)-\frac{1}{\sec(x)}] ... Start with the given expression



*[Tex \LARGE \frac{1}{\cos(x)}-\frac{1}{\frac{1}{\cos(x)}}] ... Replace each secant with {{{1/cos(x)}}}



*[Tex \LARGE \frac{1}{\cos(x)}-\cos(x)] Multiply the second fraction by the reciprocal



*[Tex \LARGE \frac{1}{\cos(x)}-\frac{\cos(x)\cdot\cos(x)}{\cos(x)}] ... Multiply the second term by {{{cos(x)/cos(x)}}}



*[Tex \LARGE \frac{1}{\cos(x)}-\frac{\cos^2(x)}{\cos(x)}] Multiply



*[Tex \LARGE \frac{1-\cos^2(x)}{\cos(x)}] Subtract the fractions



*[Tex \LARGE \frac{\sin^2(x)}{\cos(x)}] Replace *[Tex \LARGE 1-\cos^2(x)] with *[Tex \LARGE \sin^2(x)]




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




Note: you can rewrite *[Tex \LARGE \frac{\sin^2(x)}{\cos(x)}] as *[Tex \LARGE \sin(x)\frac{\sin(x)}{\cos(x)}] and then rewrite as *[Tex \LARGE \sin(x)\tan(x)] (using the identity {{{tan(x)=sin(x)/cos(x)}}})



So this also means that *[Tex \LARGE \sec(x)-\frac{1}{\sec(x)}=\sin(x)\tan(x)]