Question

Perform the subtraction and use the fundamental identities to simplify. There is more than one correct form of the answer.

Answer 1

Let's first subtract the two expressions. Use the butterfly method to subtract. The steps are:

1. Multiply the numerator of the first term by the denominator of the second term.

`(tanx)*(tanx)=tan^2x`

2. Multiply the numerator of the second term by the denominator of the first term.

`1* sec^2x=sec^2x`

3. Subtract the results in steps 1 and 2.

`tan^2x-sec^2x`

4. Divide step 3 by the product of the two denominators.

`(tan^2x-sec^2x)/(tanx)`

We have finished subtracting the two terms. Let's now simplify the difference.

Recall that one of the Pythagorean identities is 1 + tan²x = sec²x. This identity can also be written as tan²x - sec²x = -1.

Since tan²x - sec²x = -1, the numerator of the difference is just equal to -1. The expression then becomes:

`(-1)/(tanx)\text{ }or\text{ }just-(1)/(tanx)`

ANSWER:

The answer is -1/tan x.