Question

Prove that cot x sec^4 x = cot x + 2 tan x + tan^3 x

Answer 1

\[=\frac{ \cos x }{ \sin x }*\frac{ 1 }{ (\cos x)^{4} }=\frac{ 1 }{ \sin x(\cos x)^{3} }\]

Answer 2

that above is the left side.

Answer 3

right side:
\[=\frac{ \cos x }{ \sin x }+2\frac{ \sin x }{ \cos x }+\frac{ (\sin x)^{3} }{ (\cos x)^{3} }=\]
\[=\frac{ (\cos x)^{4} +2(\sin x)^{2}(\cos x)^{2}+(\sin x)^{4}}{ \sin x(\cos x)^{3} }=\]
\[=\frac{ [(\sin x)^{2}+(\cos x)^{2}]^{2} }{ \sin x(\cos x)^{3} }=\]
\[=\frac{ 1 }{ \sin x(\cos x)^{3} }\]

Answer 4

Oh wow!, thanks I forgot to factor the top.