Sinxtanx = sec x

To prove the equation sin(x)tan(x) = sec(x), we need to use trigonometric identities.

Starting with the left-hand side (LHS), we have sin(x)tan(x).

Recall that tan(x) is equal to sin(x)/cos(x). So we can rewrite the LHS as sin(x) * (sin(x)/cos(x)).

Next, we can simplify the expression by multiplying sin(x) through the parentheses:
(sin(x) * sin(x))/cos(x).

Using the identity sin^2(x) + cos^2(x) = 1, we can rewrite sin^2(x) as 1 - cos^2(x). Plugging this into the expression, we have:
((1 - cos^2(x))/cos(x)).

Now, to simplify further, we can divide both terms by cos(x):
(1/cos(x) - cos^2(x)/cos(x)).

Recall that sec(x) is equal to 1/cos(x). So we can rewrite the expression as:
sec(x) - cos^2(x)/cos(x).

Finally, simplify the expression by dividing cos^2(x) by cos(x):
sec(x) - cos(x).

This is equal to sec(x), which means sin(x)tan(x) is indeed equal to sec(x).