Prove identity.

Sec(pi-x)=-sec x

sec (π - x) = -sec (x)

when we prove, we only manipulate the left side.
first, recall that sec (x) = 1/cos x. thus we can replace the the term on the left side of equation by:
1 / cos (π - x)
then recall the sum/difference identity for cosine:
cos(A−B) = cos A cos B + sin A sin B
thus we use this to expand the term in the denominator:
1 / cos (π - x)
1 / [ cos(π)cos(x) + sin(π)sin(x) ]
note that sin (π) = 0, thus we can cancel it. also cos(π) = -1. therefore:
1 / cos(π)cos(x)
1 / -cos(x)
using the first identity we did earlier,
1/ - cos(x) = - sec(x)

hope this helps~ :)

pi -x and x have the same reference angle (x), but are in quadrants on opposite sides of the x axis. Thus the absolute values of sec x and sec (pi -x) are the same, but the signs are opposite.

q.e.d.

Oh okay thanks a lot:) I was distributing sec and couldn't' get anywhere. I didn't think of writing it like that.

Gamed

Well, let me take a hilarious approach to proving this identity.

Imagine you are at a circus, and two clowns are playing a game of tug-of-war with a rubber chicken. One clown is pulling the chicken towards him, while the other clown is pulling it away. The force applied by the first clown is represented by "sec x" and the force applied by the second clown is represented by "sec(pi-x)."

Now, if the two clowns have equal strength, the rubber chicken won't move, right? So, we can say that if "sec x" is equal to "sec(pi-x)," then no clown wins the tug-of-war, and the chicken stays put.

Using the fact that sec(theta) = 1/cos(theta), we can rewrite the equation as:

1/cos(x) = 1/cos(pi-x).

To make things simpler, let's multiply both sides of the equation by cos(x) * cos(pi-x):

1 = cos(x) * cos(pi-x).

Now, let's explore the unit circle for a moment. At any given theta, the x-value of a point on the unit circle is represented by cos(theta). So, let's plug in theta = x and theta = pi-x:

1 = cos(x) * cos(pi-x).

Now, the x-values on the unit circle for angles x and pi-x are the same. So, no matter which clowns are pulling the rubber chicken, they are exerting an equal force, and the chicken doesn't move.

Therefore, we have successfully proven that Sec(pi-x) = -sec(x) using the power of comedy and a silly circus analogy.

To prove the identity `sec(π - x) = -sec(x)`, we can use the definitions of secant and the properties of trigonometric functions.

Let's begin by expressing the left-hand side (LHS) of the equation:

sec(π - x)

Using the reciprocal identity, we have:

sec(π - x) = 1 / cos(π - x)

Since cosine is an even function, we can rewrite it as:

cos(π - x) = cos(x)

Substituting this back into the equation:

sec(π - x) = 1 / cos(x)

Now, let's simplify the right-hand side (RHS) of the equation:

-sec(x)

Using the definition of secant:

-sec(x) = -1 / cos(x)

Comparing the LHS and RHS, we see that they are equal.

Therefore, we have proved the identity `sec(π - x) = -sec(x)`.

Some key steps to remember when proving trigonometric identities:
1. Familiarize yourself with the basic trigonometric identities.
2. Rewrite the expressions on both sides of the equation using these identities.
3. Simplify both sides as much as possible.
4. Compare the simplified expressions on both sides to check for equality.

Note: It's important to remember that identities can be proven using various approaches, so there might be alternative methods to prove the same identity.