Factor and use trig identities to simplify 4 sec3x 4 sec2x 4

Factor and use trig identities to simplify

4 sec³x 4 sec²x 4 sec x + 4

Solution

4sec³x 4sec²x 4secx + 4

==> 4[sec³x sec²x secx + 1]

==> 4[ sec²x(secx 1) 1(secx - 1) ]

==> 4[ (sec²x - 1)(secx - 1) ]     ; sec²x - tan²x = 1 ==> sec²x - 1 = tan²x

==> 4 tan²x (secx -1)

Hence 4sec³x 4sec²x 4secx + 4 = 4 tan²x (secx -1)