Perform the addition or subtraction and use the fundamental identities to simplify.

tan x - sec^2 x / tan x

I do not know where to start.

The way you typed it ....

tan x - sec^2 x / tan x
= tanx - (1/cos^2x)(c0sx/sinx+
= tanx - 1/(sinxcosx)
= tanx - cscxsecx

Hannah,

if you continue by substituting sin and cos for the tan,csc and sec functions, you can simplify further. This will be left as an exercise for yourself.

To simplify the expression tan x - sec^2 x / tan x, we can use the fact that sec^2 x = 1 + tan^2 x, which is one of the fundamental trigonometric identities.

Let's simplify step by step:

1. Write the expression:
tan x - sec^2 x / tan x

2. Replace sec^2 x with 1 + tan^2 x:
tan x - (1 + tan^2 x) / tan x

3. Simplify the denominator by applying the distributive property:
tan x - 1/tan x - (tan^2 x)/tan x

4. Combine the terms in the numerator:
tan x - 1/tan x - tan x

5. Simplify further:
-1/tan x

So, the simplified expression is -1/tan x.

To summarize:

tan x - sec^2 x / tan x
= tan x - (1 + tan^2 x) / tan x
= tan x - 1/tan x - (tan^2 x)/tan x
= tan x - 1/tan x - tan x
= -1/tan x

Therefore, the simplified expression is -1/tan x.