Start from the left-hand side:
4 sec x – 4 cos x
Secant is the reciprocal of cosine:
1
= 4 · ———— – 4 cos x
cos x
4
= ———— – 4 cos x
cos x
Reduce all terms to the same denominator:
4 4 cos² x
= ———— – ——————
cos x cos x
4 – 4 cos² x
= ————————
cos x
Factor out 4 in the numerator:
4 · ( 1 – cos² x)
= ————————— but 1 – cos² x = sin² x
cos x
4 · sin² x
= ——————
cos x
4 · sin x · sin x
= —————————
cos x
sin x
= 4 · sin x · ————
cos x
= 4 · sin x · tan x ✔
I hope this helps. =)
