Prove identity.

Question

Sec(pi-x)=-sec x

Answer 1

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~ :)

Answer 2

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

Answer 3

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.

Answer 4

Gamed