SOLUTION: where x represents the theta prove that sec x(1-sin x)(sec x+tan x) = 1

Algebra ->  Trigonometry-basics -> SOLUTION: where x represents the theta prove that sec x(1-sin x)(sec x+tan x) = 1      Log On


   

Question 511681: where x represents the theta
prove that sec x(1-sin x)(sec x+tan x) = 1
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
In this proof you can make use of three identities as follows:
.
cos%28theta%29%2Asec%28theta%29+=+1 which is equivalent to sec%28theta%29+=+1%2Fcos%28theta%29
.
tan%28theta%29+=+sin%28theta%29%2Fcos%28theta%29
.
and
.
sin%5E2%28theta%29%2B+cos%5E2%28theta%29+=+1 which is equivalent to cos%5E2%28theta%29+=+1+-+sin%5E2%28theta%29
.
Start with:
.
%28sec%28theta%29%29%281-sin%28theta%29%29%28sec%28theta%29%2Btan%28theta%29%29+=+1
.
On the left side replace sec%28theta%29 by its equivalent 1%2Fcos%28theta%29 in two places to get:
.
%281%2Fcos%28theta%29%29%281-sin%28theta%29%29%281%2Fcos%28theta%29%2B+tan%28theta%29%29=1
.
Next replace tan%28theta%29 by its equivalent sin%28theta%29%2Fcos%28theta%29:
.

.
Multiply the first two terms in parentheses and the equation becomes:
.

.
In the second set of parentheses put everything over the common denominator and this makes the equation become:
.

.
Multiply the two numerators together. Also multiply the two denominators together:
.
%28%281-sin%5E2%28theta%29%29%2Fcos%5E2%28theta%29%29=1
.
Now replace the numerator by its equivalent cos%5E2%28theta%29 and the equation reduces to:
.
%28cos%5E2%28theta%29%2Fcos%5E2%28theta%29%29=1
.
Dividing out the left side further reduces the equation to:
.
1+=+1
.
And since the left side has been reduced until it equaled the right side, the proof has been validated. QED (thus it is demonstrated).
.
Hope this helps to familiarize you with some of the trig identities and how to use them.