Question

Use the fundamental identities to fully simplify the expression.

Answer 1

Answer:

First, remember that:

$tan(x) = \frac{sin(x)}{cos(x)}$

and:

$sec(x) = \frac{1}{cos(x)}$

Then we can rewrite the expression:

$tan(x)*sin(x) + sec(x)*cos^2(x) = A$

Where A is an equivalent expression.

as:

$\frac{sin(x)}{cos(x)} *sin(x) + \frac{1}{cos(x)}*cos^2(x) = A$

Then this is:

$\frac{sin^2(x)}{cos(x)} + cos(x) = A$

Now we can multiply both sides by cos(x), then:

$(\frac{sin^2(x)}{cos(x)} + cos(x))*cos(x) = A*cos(x)$

$sin^2(x) + cos^2(x) = A*cos(x)$

And we know that the left term of the above equation is equal to 1, then:

$1 = A*cos(x)$

$\frac{1}{cos(x)} = sec(x) = A$

And A is equivalent to the original expression, then we get:

$tan(x)*sin(x) + sec(x)*cos^2(x) = sec(x)$

The simplification of the expression is sec(x)