How do you simplify (tanx + 1)^2 cosx?
1 Answer
Explanation:
Let's begin by expanding the bracket.
(tanx+1)^2=tan^2x+2tanx+1
color(orange)"Reminder " color(red)(bar(ul(|color(white)(a/a)color(black)(tanx=(sinx)/(cosx))color(white)(a/a)|)))
rArrtan^2x+2tanx+1=(sin^2x)/(cos^2x)+(2sinx)/cosx+1 substitute this back into the original.
That is
(sin^2x/cos^2x+(2sinx)/cosx+1)cosx distribute the bracket.
rArrsin^2x/cosx+2sinx+cosx express each term with a common denominator of cosx
=(sin^2x+2sinxcosx+cos^2x)/(cosx)........ (A)
color(orange)"Reminder " color(red)(bar(ul(|color(white)(a/a)color(black)(sin^2x+cos^2x=1)|))) and
color(red)(bar(ul(|color(white)(a/a)color(black)(sin2x=2sinxcosx)color(white)(a/a)|))) Returning to (A)
rArr(1+sin2x)/cosx=1/cosx+(sin2x)/cosx
color(orange)"Reminder " color(red)(bar(ul(|color(white)(a/a)color(black)(secx=1/(cosx))color(white)(a/a)|)))
rArr1/cosx+(sin2x)/cosx=secx+secxsin2x
rArr(tanx+1)^2cosx=secx(1+sin2x)
