Question

If someone could help i really need this... verify the identities, (sin x+tan x+ tan^3 x)/tan x= cos x + sec^2 x

Answer 1

Start with the left side, and remember that tan x = sin x / cos x:\[\frac{ \sin x+\tan x+\tan ^3x }{ \tan x }=\]\[\frac{ \sin x+\frac{ \sin x }{ \cos x }+\frac{ \sin ^{3}x }{ \cos ^{3}x } }{ \frac{ \sin x }{ \cos x } }=\]\[ \left( \sin x+\frac{ \sin x }{ \cos x }+\frac{ \sin ^{3}x }{ \cos ^{3}x } \right)*\frac{ \cos x }{ \sin x }=\]\[\cos x+1+\frac{ \sin ^{2}x }{ \cos ^{2}x }=\]\[\cos x+1+\tan ^{2}x=\]\[\cos x+\sec ^{2}x\]
And we have verified the equality.
If there is anything unclear, please ask, I'll try to explain.

ZeHanz

Answer 2

Well, I like doing these kind of probs, but one wonders sometimes: how urgent is it?