Question

Simplify and write the trigonometric expression in terms of sine and cosine: sec t − cos t sec t = ( f ( t ) ) 2 sect-costsect=(f(t))2

Answer 1

`f(t) = 1 - cos^2\ t` and
`f(t) = sin^2\ t`

Explanation:

Given

`(sec\ t - cos\ t)/(sec\ t) = f(t)`

Required

Simplify

`f(t) = (sec\ t - cos\ t)/(sec\ t)`

Split fraction

`f(t) = (sec\ t )/(sec\ t) - (cos\ t)/(sec\ t)`

`f(t) = 1 - (cos\ t)/(sec\ t)`

`sec\ t = (1)/(cos\ t)`

So, we have:

`f(t) = 1 - (cos\ t)/((1)/(cos\ t))`

`f(t) = 1 - cos\ t * cos\ t`

`f(t) = 1 - cos^2\ t`

In trigonometry:

`1 - cos^2\ t = sin^2\ t`

So, we have:

`f(t) = sin^2\ t`