Question

Verify the identity: 4secx-4cosx=4sinxtanx

Answer 1

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. =)