Prove the identity: $\dfrac{\sin x + \cos x}{\cos^3 x} = \tan^3 x + \tan^2 x + \tan x + 1$
$\begin{aligned}
RHS & = \tan^3 x + \tan^2 x + \tan x + 1 \\\\
& = \tan^2 x \left(\tan x + 1\right) + 1 \left(\tan x + 1\right) \\\\
& = \left(1 + \tan^2 x\right) \left(1 + \tan x\right) \\\\
& = \sec^2 x \left(1 + \tan x\right) \\\\
& = \dfrac{1}{\cos^2 x} \left(1 + \dfrac{\sin x}{\cos x}\right) \\\\
& = \dfrac{\sin x + \cos x}{\cos^3 x} = LHS
\end{aligned}$
Hence proved.