Question

standard 16 homework
score: 4/23 answered: 3/16
question 4
simplify the following expression to a single trig function with no fractions.
(sec(t) - cos(t)) / tan(t) =
question help:

Explanation:

Step1: Replace sec(t) with 1/cos(t)

We know that the secant function is the reciprocal of the cosine function, so $\sec(t)=\frac{1}{\cos(t)}$. Substituting this into the numerator, we get:
$\frac{\frac{1}{\cos(t)} - \cos(t)}{\tan(t)}$

Step2: Simplify the numerator

To simplify the numerator, we find a common denominator, which is $\cos(t)$. So:
$\frac{1 - \cos^{2}(t)}{\cos(t)}$
And by the Pythagorean identity, $1 - \cos^{2}(t)=\sin^{2}(t)$. So the numerator becomes $\frac{\sin^{2}(t)}{\cos(t)}$

Step3: Replace tan(t) with sin(t)/cos(t)

The tangent function is $\tan(t)=\frac{\sin(t)}{\cos(t)}$. So now our expression is:
$\frac{\frac{\sin^{2}(t)}{\cos(t)}}{\frac{\sin(t)}{\cos(t)}}$

Step4: Divide the fractions

Dividing by a fraction is the same as multiplying by its reciprocal. So we have:
$\frac{\sin^{2}(t)}{\cos(t)} \times \frac{\cos(t)}{\sin(t)}$

Step5: Simplify the expression

The $\cos(t)$ terms cancel out, and we have $\sin^{2}(t)/\sin(t)=\sin(t)$

Answer:

$\sin(t)$