Simplify sec x - (cos x divided by sin x + 1)

Question

answer is tan x but how do you get there step by step?

Answer 1

IF you think tanx is the solution, try a couple angles and see if it's equivalent to secx-(cosx/sinx +1)

try 45 degrees

tan45 = 1

sec45 = sqr2

cos45/sin45 = cot45 = 1

sec45-(cot45+1) = sqr2 - 2 = 1.414-2 = about -.586 which doesn't = tan45 = 1

maybe the problems was supposed to be secx-(cosx/(sinx+1)) ? Try an angle, such as 45 degrees

sqr2 - (1/sqr2/(1/sqr2+1) = 1.414 - (.707/1.707) = 1.414-.414 = 1. That works, so there must be a

missing parentheses: secx-(cosx/(sinx+1)) = tanx

secx -(cosx/(sinx+1) =

1/cosx - cosx/(sinx+1) (multiply 2nd part, the fraction by (sinx+1)/(sinx+1) = 1 In other words multiply the denominator by its conjugate, and multiply the numerator by that same conjugate of the denominator to get

the following which has sin^2x-1 in the denominator. sin^2x -1 = -cos^2

1/cosx - (cosxsinx + cosx)/(sin^2 - 1) = 1/cosx - (cosxsinx - cosx)/(-cos^2x) then cancel cosx from numerator and dernominator

1/cosx + (sinx - 1)/cosx combine both fractions over the denominator cosx

(1+sinx-1)/cosx = sinx/cosx = tanx