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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

Sagot :

MrRoyalidn

Answer:

[tex]f(t) = 1 - cos^2\ t[/tex] and [tex]f(t) = sin^2\ t[/tex]

Step-by-step explanation:

Given

[tex]\frac{sec\ t - cos\ t}{sec\ t} = f(t)[/tex]

Required

Simplify

[tex]f(t) = \frac{sec\ t - cos\ t}{sec\ t}[/tex]

Split fraction

[tex]f(t) = \frac{sec\ t }{sec\ t} - \frac{cos\ t}{sec\ t}[/tex]

[tex]f(t) = 1 - \frac{cos\ t}{sec\ t}[/tex]

[tex]sec\ t = \frac{1}{cos\ t}[/tex]

So, we have:

[tex]f(t) = 1 - \frac{cos\ t}{\frac{1}{cos\ t}}[/tex]

[tex]f(t) = 1 - cos\ t * cos\ t[/tex]

[tex]f(t) = 1 - cos^2\ t[/tex]

In trigonometry:

[tex]1 - cos^2\ t = sin^2\ t[/tex]

So, we have:

[tex]f(t) = sin^2\ t[/tex]

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