Answer:
First, remember that:
[tex]tan(x) = \frac{sin(x)}{cos(x)}[/tex]
and:
[tex]sec(x) = \frac{1}{cos(x)}[/tex]
Then we can rewrite the expression:
[tex]tan(x)*sin(x) + sec(x)*cos^2(x) = A[/tex]
Where A is an equivalent expression.
as:
[tex]\frac{sin(x)}{cos(x)} *sin(x) + \frac{1}{cos(x)}*cos^2(x) = A[/tex]
Then this is:
[tex]\frac{sin^2(x)}{cos(x)} + cos(x) = A[/tex]
Now we can multiply both sides by cos(x), then:
[tex](\frac{sin^2(x)}{cos(x)} + cos(x))*cos(x) = A*cos(x)[/tex]
[tex]sin^2(x) + cos^2(x) = A*cos(x)[/tex]
And we know that the left term of the above equation is equal to 1, then:
[tex]1 = A*cos(x)[/tex]
[tex]\frac{1}{cos(x)} = sec(x) = A[/tex]
And A is equivalent to the original expression, then we get:
[tex]tan(x)*sin(x) + sec(x)*cos^2(x) = sec(x)[/tex]
The simplification of the expression is sec(x)
