How do you perform addition or subtraction and use the fundamental identities to simplify #1/(secx+1)-1/(secx-1)#?

To simplify the expression 1/(sec(x) + 1) - 1/(sec(x) - 1) using fundamental trigonometric identities, we start by rewriting sec(x) in terms of cosine.

Since sec(x) is the reciprocal of cosine, we have sec(x) = 1/cos(x).

Next, we'll use the common denominator approach to combine the fractions:

1/(sec(x) + 1) - 1/(sec(x) - 1) = 1/(1/cos(x) + 1) - 1/(1/cos(x) - 1) = cos(x)/(cos(x) + 1) - cos(x)/(cos(x) - 1)

Now, to combine the fractions, we find a common denominator, which is the product of the denominators:

Common denominator = (cos(x) + 1)(cos(x) - 1)

Now, rewrite each fraction with the common denominator:

= [cos(x)(cos(x) - 1)] / [(cos(x) + 1)(cos(x) - 1)] - [cos(x)(cos(x) + 1)] / [(cos(x) + 1)(cos(x) - 1)]

Next, simplify each fraction:

= [cos^2(x) - cos(x)] / [cos^2(x) - 1] - [cos^2(x) + cos(x)] / [cos^2(x) - 1]

Now, subtract the fractions:

= [cos^2(x) - cos(x) - cos^2(x) - cos(x)] / [cos^2(x) - 1]

Combine like terms:

= [-2cos(x)] / [cos^2(x) - 1]

Finally, we can use the fundamental trigonometric identity cos^2(x) - 1 = -sin^2(x):

= [-2cos(x)] / [-sin^2(x)]

Divide both the numerator and the denominator by -1:

= [2cos(x)] / [sin^2(x)]

Thus, the simplified expression for 1/(sec(x) + 1) - 1/(sec(x) - 1) using fundamental trigonometric identities is 2cos(x) / sin^2(x).