Question

trigonometry
3) Simplify.
a)sec x/tan x
b)(sec - 1) (sec x + 1)

Answer 1

The first one just write sec x as 1/cosx and write tan x as sinx/cosx

So now just divide 1/ cos(x) by sin(x)/ cos(x)

So invert and multiply.
1/ cos(x) * cos(x)/sin(x) = 1/sin(x)
1/sin(x) = csc(x)

For b, note you have a difference of squares. So (sec(x) -1)(sec(x)+1) = sec^2(x) -1

Write sec^2(x) as 1/cos^2(x) So now you have this:

1/cos^2(x) - 1

Find a common denominator and rewrite equivalent fractions.

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

Note that 1-cos^2(x) is the Pythagorean identity and equals sin^2(x)

So we now have this:

sin^2(x)/cos^2(x) = tan^2(x)