Rewrite Expression

Question

Rewrite the expression in terms of sines and cosines only and then simplify the resulting expression.

 

tanx/secx+1 +tan x/sec x-1

Answer 1

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

Multiply numerator and denominator of each term by cos x:

sinx/(1+cosx) + sin(x)/(1-cosx)

Multiply first term by (1-cosx) and 2nd term by (1+cosx) -- both numerator and denominator in order to get a common denominator:

[sinx(1-cosx) + sinx(1+cosx)]/(1 - cos²x)

Factor sinx in the numerator and rewrite the denominator as sin²x. Then cancel the sinx from the numerator:

[1 - cosx + 1 + cosx]/sinx

Resulting in 2/sinx or 2 cscx.