Answer
$(\tan x+\cot x)^{2}=\sec^{2}x+\csc^{2}x$
Work Step by Step
$(\tan x+\cot x)^{2}=\sec^{2}x+\csc^{2}x$
Evaluate the power on the left side of the equation:
$\tan^{2}x+2\tan x\cot x+\cot^{2}x=\sec^{2}x+\csc^{2}x$
Substitute $\tan x$ with $\dfrac{\sin x}{\cos x}$ and $\cot x$ with $\dfrac{\cos x}{\sin x}$:
$\tan^{2}x+\cot^{2}x+2\Big(\dfrac{\sin x}{\cos x}\Big)\Big(\dfrac{\cos x}{\sin x}\Big)=\sec^{2}x+\csc^{2}x$
$\tan^{2}x+\cot^{2}x+2=\sec^{2}x+\csc^{2}x$
Substitute $\tan^{2}x$ with $\sec^{2}x-1$ and $\cot^{2}x$ with $\csc^{2}x-1$:
$\sec^{2}x-1+\csc^{2}x-1+2=\sec^{2}x+\csc^{2}x$
Simplify and finally the identity is proved:
$\sec^{2}x+\csc^{2}x=\sec^{2}x+\csc^{2}x$
