Rewrite the expression so that it is not in fractional form: (cos x)/(sec x - 1)?

Answer 1

First rewrite and cos(x)/(1/cos(x)-1)
Which equals cos^2(x)/(1-sin(x))
Which equals (1-sin(x))(1+sin(x))/(1-sin(x))
Which equals 1+sin(x)

Answer 2

Multiply numerator and denominator by (sec x + 1) to get...

(1 + cos x)/(sec^2 - 1)

Replace denominator with tan^2 x

(1 + cos x)/(tan^2 x)

Write write tan^2 x as the reciprocal of cot ^2 x

(1 + cos x)(cot^2 x)

Answer 3

It's (cot^2 x) (1 + cos x). Here's how this is obtained:

first multiply both numerator and denominator by cos x; then

(cos x)/(sec x - 1) = (cos^2 x)/(1 - cos x). ......(A)

But 1 - cos^2 x = (1 - cos x) (1 + cos x) = sin^2 x, so that

1/(1 - cos x) = (1 + cos x)/(sin^2 x). Therefore expression (A) equals

(cos^2 x)[(1 + cos x)/(sin^2 x)] = (cot^2 x)(1 + cos x).

QED

Live long and prosper.

Answer 4

sec x = 1/cos x

Therefore it can be written as 1-cos x

Answer 5

sec x - 1 = (1/sin x )- 1 = (1-sin x) /( sin x)
(cos x)/((1-sin x)/(sin x))
multiply top and bottom by (sin x)
so,
(cos x ) (sin x)/ (1 - sin x)
multiply top and bottom by (1+sin x)
(cos x) (sin x) (1 + sin x) / (1 - sin^2 x)
1 - sin^2 x = cos^2 x
substituting you get
(cos x)(sin x)/ (cos^2 x) and a cos x cancels and you get
(sin x)/(cos x) = tan x