tan^4k -sec^4k=1-2sec^2k, need to show how this is an identity?

Answer 1

it can be done simpler

(tan ^ 4 k - sec^4 k =(tan^2 k - sec^2 k)(tan^2 k + sec^2 k)
(using the fomula (a^2-b^2) = (a+b)(a-b) a = tan^2 k b = sec^2 k)
= -1(sec^2 k-1+ sec^2 k) as tan^2 k +1 = sec^2 k
= 1-2 sec^2 k

Answer 2

(tan k)^4 - (sec k)^4

= ((sec k)^2 - 1)^2 - (sec k)^4

I think you can finish it yourself from there -- just expand the parentheses and collect terms.

Answer 3

tanK^4 - secK^4
(tanK^2)^2 - (secK^2)^2
(secK^2 - 1)^2 - (secK^2)^2
((1/cosK)^2 - 1)^2 - ((1/cosK)^2)^2
((1 - cosK^2)/(cosK^2))^2 - ((1/cosK)^2)^2

((1 - cosK^2)^2 - 1)/((cosK)^4)

(((1 - cosK^2)(1 - cosK^2)) - 1)/(cosK^4)

((1 - cosK^2 - cosK^2 + cosK^4) - 1)/(cosK^4)

(1 - 2cosK^2 + cosK^4 - 1)/(cosK^4)

(cosK^4 - 2cosK^2)/(cosK^4)

((cosK^2)(cosK^2 - 2))/(cosK^4)

(cosK^2 - 2)/(cosK^2)

(cosK^2 / cosK^2) - (2/cosK^2)

1 - 2secK^2

so therefore

tanK^4 - secK^4 = 1 - 2secK^2