+466 (sec3 x - 8)/(sec2 x + sec x - 6) = (sec2 x + 2 sec x + 4)/(sec x + 3)
Verify the following identity:
(sec(x)^3-8)/(sec(x)^2+sec(x)-6) = (sec(x)^2+2 sec(x)+4)/(sec(x)+3)
Cancel sec(x)-2 from the numerator and denominator. (sec(x)^3-8)/(-6+sec(x)+sec(x)^2) = ((sec(x)-2) (4+2 sec(x)+sec(x)^2))/((sec(x)-2) (3+sec(x))) = (4+2 sec(x)+sec(x)^2)/(3+sec(x)):
(4+2 sec(x)+sec(x)^2)/(3+sec(x)) = ^?(4+2 sec(x)+sec(x)^2)/(3+sec(x))
The left hand side and right hand side are identical:
Answer: | (identity has been verified)
+129852 (sec^3 x - 8)/(sec^2 x + sec x - 6) = (sec^2 x + 2 sec x + 4)/(sec x + 3)
Factor (sec^3x- 8) as a difference of cubes =
(secx - 2) (sec^2x + 2secx + 4)
Factor (sec^2 x + sec x - 6) as ( secx + 3) (secx - 2)
So we have
[ (secx - 2) (sec^2x + 2secx + 4)] / [ ( secx + 3) (secx - 2) ]
[secx - 2] cancels on top/bottom....and we're left with
(sec^2x + 2secx + 4) / ( secx + 3) = the right hand side
