Question

How do i verify sec x- cos x = sin x tan x?

Answer 1

See below.

Explanation:

When verifying these identities, it's best to leave one side alone, and work with the more complex side to get it in the same form as the side left alone.

Here, the left side appears more complex, so we'll work with that.

Convert things to be in terms of sine and cosine. Recall that `secx=1/cosx:`

`1/cosx-cosx=sinxtanx`

Subtract the expressions on the left, using `cosx` as a common denominator.

`(1-cos^2x)/cosx=sinxtanx`

Now, recall the Pythagorean identity `sin^2x+cos^2x=1.` This identity also tells us that `1-cos^2x=sin^2x.`

`sin^2x/cosx=sinxtanx`

Split up the square:

`(sinxsinx)/cosx=sinxtanx`

Recalling that `sinx/cosx=tanx:`

`sinxtanx=sinxtanx`

So, the identity is verified.