prove the identities?

Question

cos(x + π/2) = -sinx

cos^4x - sin^4x = cos2x

sinx secx = tanx

Answer 1

cos(x + π/2) =
cos x cos π/2 - sin x sin π/2 =
cos x (0) - sin x (1) =
-sin x

cos^4 x - sin^4 x =
(cos² x + sin² x)(cos² x - sin² x) =
(1)(cos² x - sin² x) =
cos 2x

sin x sec x =
sin x (1/cos x) =
(sin x)/(cos x) =
tan x

Answer 2

Quite simple, actually:

cos (x+π/2) = cos x cos (π/2) - sin x sin (π/2) = cos x * 0 - sin x * 1 = -sin x

cos⁴ x - sin⁴ x = (cos² x - sin² x)(cos² x + sin² x) = (cos (2x))(1) = cos (2x)

sin x sec x = sin x * 1/cos x = tan x