How do you simplify the expression tan^2x/(secx+1)?

2 Answers
Ratnaker Mehta
Sep 1, 2016

tan^2 x=sec^2 x-1=(sec x+1)(sec x -1)

rArr tan^2 x/(sec x+1)=sec x-1, or, =(1-cos x)/cos x,

Jim G.
Sep 1, 2016

secx-1

Explanation:

Use the color(blue)"trigonometric identity" for tan^2x

color(orange)"Reminder " color(red)(|bar(ul(color(white)(a/a)color(black)(tan^2x=sec^2x-1)color(white)(a/a)|)))

rArr(tan^2x)/(secx+1)=(sec^2x-1)/(secx+1)........ (A)

Note that sec^2x-1" is a difference of squares" and in general, factorises as.

color(red)(|bar(ul(color(white)(a/a)color(black)(a^2-b^2=(a-b)(a+b))color(white)(a/a)|)))

(secx)^2=sec^2x" and " (1)^2=1rArra=secx" and " b=1

rArrsec^2x-1=(secx-1)(secx+1)

substitute into (A)

rArr(sec^2x-1)/(secx+1)=((secx-1)cancel((secx+1)))/cancel(secx+1)

rArr(tan^2x)/(secx+1)=secx-1

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