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

Answer:

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

Step-by-step explanation:

Given

$\frac{sec\ t - cos\ t}{sec\ t} = f(t)$

Required

Simplify

$f(t) = \frac{sec\ t - cos\ t}{sec\ t}$

Split fraction

$f(t) = \frac{sec\ t }{sec\ t} - \frac{cos\ t}{sec\ t}$

$f(t) = 1 - \frac{cos\ t}{sec\ t}$

$sec\ t = \frac{1}{cos\ t}$

So, we have:

$f(t) = 1 - \frac{cos\ t}{\frac{1}{cos\ t}}$

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

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

In trigonometry:

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

So, we have:

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