For 0° < θ < 90°, the least value of tanθ + 4cotθ is:

Put $t$ = tanθ, which is positive on this interval, so the expression is $t+4/t$. By the AM–GM inequality $t+4/t$ ≥ 2√$(t\times 4/t)=2\times 2=4$, with equality when $t=4/t$, i.e. $t=2$ (θ ≈ 63.4°), where the value really is 2 + 2 = 4. Quoting the familiar tanθ + cotθ ≥ 2 forgets the coefficient 4 and gives 2; adding the coefficients 1 and 4 gives 5; and doubling the correct bound gives 8, which the value 4 at θ ≈ 63.4° already contradicts.