Question

I'm at the very end of a calculus problem where I'm supposed to use a cofunction identity to get the final result, but I literally am just misapplying the cofunction identity and need somebody to check up on this. Posted below in a minute.

Answer 1

\[-2\csc( \frac{x - \pi }{2})\]

According to W|A this is supposed to result in sec(x/2), but somehow I'm butchering my understanding of the cofunction identities enough for that not to be the case. Here's what I'm doing:

Answer 2

Because\[\csc(u) = \sec( \frac{\pi}{2}-u),\]\[-2\csc( \frac{x - \pi }{2}) = -2\sec( \frac {\pi}{2}-[ \frac{x - \pi }{2}]) = -2\sec(\pi - \frac{x}{2})\]

Answer 3

Oh, and it's 2sec(x/2), not sec(x/2), but that shouldn't really matter, apparently; I think they're using an odd-even identity to commute in the (-1) on the csc to reverse the argument therefore causing the argument of sec(pi/2-u) to change; I think that's it, lol, I might've just resolved what I was missing.

Answer 4

\[2\csc(-( \frac {x- \pi}{2}) = 2\csc( \frac {\pi - x}{2}) =2\sec ( \frac {\pi}{2} - ( \frac{\pi - x}{2})) = 2\sec( \frac{x}{2})\]

Answer 5

you are doing good job of working this out on your own :)
\[\sec(\pi - A) = -\sec(A)\]
using difference identity for cosine

Answer 6

Wait, so my approach was also valid, right, lol.

Answer 7

yep

Answer 8

Thanks!