[tex]\sec\left(-\dfrac\pi{12}\right)[/tex]
First use the fact that [tex]\sec x[/tex] is an even function, i.e. [tex]\sec(-x)=\sec x[/tex]. So this is the same as
[tex]\sec\dfrac\pi{12}[/tex]
Now, note that [tex]\dfrac\pi{12}=\dfrac\pi3-\dfrac\pi4[/tex], so
[tex]\sec\dfrac\pi{12}=\sec\left(\dfrac\pi3-\dfrac\pi4\right)=\dfrac1{\cos\frac\pi3\cos\frac\pi4+\sin\frac\pi3\sin\frac\pi4}[/tex]
which comes from the sum identity for cosine,
[tex]\cos(x\pm y)=\cos x\cos y\mp\sin x\sin y[/tex]
Now,
[tex]\sec\dfrac\pi{12}=\dfrac1{\frac12\frac1{\sqrt2}+\frac{\sqrt3}2\frac1{\sqrt2}}=\sqrt6-\sqrt2[/tex]
