Expert: Sherman D. - 2/23/2005

Question
simplify the expressions.

1) (sec x + tan x)(1 - sin x)
2) cot x + tan x / csc^2 x

Answer
1.)
(sec(x) + tan(x))(1 - sin(x))
sec(x) = 1/(cos(x))
tan(x) = (sin(x))/(cos(x))

((1/(cos(x))) + ((sin(x))/(cos(x))) * (1 - sin(x))
((1 + sin(x))/(cos(x))) * (1 - sin(x))
(((1 + sin(x)) * (1 - sin(x)))/(cos(x))
(1 + sin(x) - sin(x) - sin(x)^2)/(cos(x))
(1 - sin(x)^2)/(cos(x))

1 - sin(x)^2 = cos(x)^2

so this gives you

(cos(x)^2)/(cos(x))
or
(cos(x) * cos(x))/(cos(x))

one set of the cos(x)'s cancel out, leaving you with

cos(x) as your answer

so

(sec(x) + tan(x))(1 - sin(x)) = cos(x)

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2.)
cot(x) + tan(x) / csc^2x

if by this you mean (cot(x) + tan(x))/(csc(x)^2)

((1/(tan(x))) + tan(x))/((1/(sin(x)))^2)
((1 + tan(x)^2)/(tan(x)) / ((1/(sin(x)^2))
((sec(x)^2))/(tan(x)) / (1/(sin(x)^2)))
((1/(cos(x))^2/((sin(x))/(cos(x))) / ((1/(sin(x))^2)
((1/(cos(x))^2 * (cos(x)/(sin(x))) / (1/(sin(x)^2)))
(cos(x)/(cos(x)^2 * sin(x))) / (1/(sin(x^2)))
(1/(cos(x) * sin(x))) / (1/(sin(x)^2)))
(1/(cos(x) * sin(x))) * ((sin(x)^2)/1)
(sin(x)^2)/(cos(x) * sin(x))
(sin(x))/(cos(x))
ANS : tan(x)