Question 1028156
Please show me how to prove:
(cscx-1)(1+cscx)=(cscx cosx)/(secx sinx)
<pre>Usually, one side is chosen and then that side is proven to be equal to the other. With this, each side needs to be altered as follows:
{{{highlight_green((csc (x) - 1)(1 + csc (x))) = highlight((csc (x) cos (x))/(sec (x) sin (x))))}}}
<b><u>LEFT side:</b></u>
{{{highlight_green((csc (x) - 1)(1 + csc (x)))}}}
{{{highlight_green((csc (x) - 1)(csc (x) + 1))}}} ------- Rearranging binomial
{{{highlight_green(csc^2 (x) - 1)}}} ------ FOILing binomials
{{{highlight_green(cot^2 (x))}}} ------ Applying IDENTITY: {{{cot^2 (x) = csc^2 (x) - 1)}}} 

<b><u>RIGHT side:</b></u>
{{{highlight((csc (x) cos (x))/(sec (x) sin (x))))}}}
{{{highlight(((1/sin (x)) * cos (x))/((1/cos (x)) * sin (x))))}}} ------ Replacing {{{highlight(matrix(1,7, csc (x), with, 1/sin (x), and, sec (x), with, 1/cos (x)))}}}
{{{highlight(matrix(1,3, cos (x)/sin (x), "÷", sin (x)/cos (x))))}}}_____{{{highlight(matrix(1,3, cos (x)/sin (x), "*", cos (x)/sin (x))))}}}_____{{{highlight(cos^2 (x)/sin^2 (x))}}} ----- {{{highlight(cot^2 (x))}}}
{{{highlight_green((csc (x) - 1)(1 + csc (x))) = highlight((csc (x) cos (x))/(sec (x) sin (x))))}}} ------> {{{highlight_green(cot^2 (x)) = highlight(cot^2 (x))}}} (PROVEN)