How do I verify that the following equations is an identity: 1+ sec x sin x tan x = sec^2 x

First change sec x sin x tan x to (sinx/cosx)*tanx

= tan^2 x
So you have
1 + tan^2 x = sec^2 x
You can write the left side as
(cos^2 x + sin^2 x)/cos^2 x = 1/cos^2 x = sec^2 x

That completes the proof

Thank you!

To verify whether the equation 1 + sec(x)sin(x)tan(x) = sec^2(x) is an identity, you need to evaluate both sides of the equation and show that they are equal for all valid values of x.

First, let's manipulate the left-hand side (LHS) of the equation using the trigonometric identities:

1 + sec(x)sin(x)tan(x)

Now, substitute tan(x) with sin(x)/cos(x), and sec(x) with 1/cos(x):

1 + (1/cos(x))(sin(x))(sin(x)/cos(x))

Next, simplify the expression:

1 + sin^2(x)/cos(x)

To combine the terms, find a common denominator:

(cos(x) + sin^2(x))/cos(x)

Now, simplify further:

(cos(x) + (1 - cos^2(x)))/cos(x)

By distributing the numerator, we get:

1/cos(x)

Which is equal to sec^2(x) - the right-hand side (RHS) of the original equation.

Thus, we have shown that the LHS = RHS for all valid values of x, meaning that the equation 1 + sec(x)sin(x)tan(x) = sec^2(x) is an identity.